🧠:<think>Alright, so I've got this problem about analyzing a Floorfiller music track using Fourier analysis. Let me try to break it down step by step. I'm a bit rusty on some of these concepts, but I'll take it slow and see if I can figure it out.First, the problem is divided into two parts. The first part is about writing the function ( f(t) ) as a sum of sinusoidal functions given some frequency components and their amplitudes. The second part is about computing the power spectral density (PSD) and the total power of the signal. Let me tackle them one by one.Part 1: Writing ( f(t) ) as a sum of sinusoidal functionsOkay, so the music reviewer captured a 10-second sample of a Floorfiller track. The function ( f(t) ) is continuous and periodic with a period of 10 seconds. It has significant frequency components at 2 Hz, 5 Hz, and 11 Hz with amplitudes 3, 7, and 2 respectively.Hmm, so I remember that any periodic function can be represented as a sum of sine and cosine functions with different frequencies and amplitudes. This is the essence of Fourier series. Since the function is periodic with period ( T = 10 ) seconds, its fundamental frequency ( f_0 ) is ( 1/T = 0.1 ) Hz. But in this case, the significant frequencies are given as 2 Hz, 5 Hz, and 11 Hz. These are likely the harmonic frequencies of the fundamental frequency.Wait, let's check if 2, 5, and 11 are integer multiples of the fundamental frequency. The fundamental frequency is 0.1 Hz, so the first harmonic would be 0.2 Hz, second at 0.3 Hz, and so on. But 2 Hz is 20 times the fundamental frequency, 5 Hz is 50 times, and 11 Hz is 110 times. So these are all integer multiples, meaning they are harmonics. That makes sense because in a periodic function, the Fourier series only contains frequencies that are integer multiples of the fundamental frequency.But wait, the problem just states that the function has significant frequency components at these Hz. It doesn't specify whether they are sine or cosine functions or if they have any phase shifts. Since it's a general function, I think we can assume it's a combination of sine and cosine functions for each frequency. However, without specific phase information, we might just represent them as a combination of sine and cosine with the given amplitudes.But hold on, in Fourier series, each harmonic is represented as a combination of sine and cosine terms with specific coefficients. However, if we don't have phase information, we can represent each frequency component as a single sinusoid with a certain amplitude and phase. But since the problem gives us the amplitudes, maybe we can write each component as a cosine function with zero phase for simplicity.Alternatively, if we consider that the Fourier series can be written in terms of sine and cosine, but without knowing the phase, we can just write each component as a cosine function with the given amplitude. So, perhaps the function ( f(t) ) can be written as the sum of cosine functions with the given frequencies and amplitudes.But wait, in the Fourier series, each term is typically written as ( A_n cos(nomega_0 t + phi_n) ), where ( omega_0 ) is the fundamental frequency in radians per second. Since we have the frequencies in Hz, we can convert them to angular frequencies by multiplying by ( 2pi ).Let me recall that ( omega = 2pi f ). So, for each frequency:- 2 Hz: ( omega_1 = 2pi times 2 = 4pi ) rad/s- 5 Hz: ( omega_2 = 2pi times 5 = 10pi ) rad/s- 11 Hz: ( omega_3 = 2pi times 11 = 22pi ) rad/sBut since the function is periodic with period 10 seconds, the fundamental frequency is ( f_0 = 1/10 = 0.1 ) Hz, so ( omega_0 = 2pi times 0.1 = 0.2pi ) rad/s.Wait, so 2 Hz is 20 times the fundamental frequency, 5 Hz is 50 times, and 11 Hz is 110 times. So, in terms of the Fourier series, these would be the 20th, 50th, and 110th harmonics.But in the Fourier series, each term is ( A_n cos(nomega_0 t + phi_n) ). So, if we have significant components at 20th, 50th, and 110th harmonics, we can write ( f(t) ) as the sum of these cosine terms with their respective amplitudes.However, the problem states that the amplitudes are 3, 7, and 2 for 2 Hz, 5 Hz, and 11 Hz respectively. So, I think we can write each term as ( A_n cos(nomega_0 t) ), assuming zero phase shift for simplicity.Therefore, the function ( f(t) ) can be written as:( f(t) = 3 cos(4pi t) + 7 cos(10pi t) + 2 cos(22pi t) )Wait, but let me double-check. Since 2 Hz is 20th harmonic, so ( n = 20 ), so ( omega_n = 20 times 0.2pi = 4pi ), which matches. Similarly, 5 Hz is 50th harmonic, so ( omega_n = 50 times 0.2pi = 10pi ), and 11 Hz is 110th harmonic, so ( omega_n = 110 times 0.2pi = 22pi ). So yes, that seems correct.Alternatively, if we consider that the function is represented as a sum of sine and cosine functions without necessarily being tied to the harmonics, but just as separate frequency components, we could also write it as:( f(t) = 3 cos(4pi t) + 7 cos(10pi t) + 2 cos(22pi t) )But I think since the function is periodic with period 10 seconds, the frequencies must be integer multiples of the fundamental frequency, which they are. So, writing them as harmonics is appropriate.Alternatively, if the function is not necessarily starting at a specific phase, we might need to include phase terms, but since the problem doesn't specify, I think it's safe to assume zero phase for simplicity.So, I think the function ( f(t) ) is:( f(t) = 3 cos(4pi t) + 7 cos(10pi t) + 2 cos(22pi t) )But wait, another thought: in Fourier series, the coefficients are usually half of the amplitude for sine and cosine terms, but in this case, the problem states the amplitudes directly. So, perhaps each term is just the amplitude times cosine, without any division by 2. So, I think my initial expression is correct.Part 2: Computing the Power Spectral Density (PSD) and Total PowerAlright, moving on to the second part. The reviewer wants to compute the PSD and determine the total power over the 10-second period.I remember that the power spectral density is a measure of a signal's power content across different frequencies. For a deterministic signal like this, which is a sum of sinusoids, the PSD will have impulses (delta functions) at the frequencies of the sinusoids, with magnitudes equal to the square of their amplitudes divided by two (since each sinusoid contributes to both positive and negative frequencies).But wait, let me recall the exact definition. For a continuous-time signal, the PSD is the magnitude squared of the Fourier transform divided by the bandwidth. However, since this is a periodic signal, its Fourier transform consists of impulses at the harmonic frequencies. The power at each frequency is given by the square of the amplitude of the corresponding Fourier coefficient divided by the period.Wait, actually, for a periodic signal, the average power is the sum of the squares of the amplitudes of the Fourier series coefficients divided by two (since each cosine term contributes half its amplitude squared to the power). But I need to be careful here.Let me think step by step.First, the Fourier series of ( f(t) ) is given by:( f(t) = sum_{n=-infty}^{infty} c_n e^{j n omega_0 t} )Where ( omega_0 = 2pi / T = 0.2pi ) rad/s.But in our case, the function is expressed as a sum of cosines:( f(t) = 3 cos(4pi t) + 7 cos(10pi t) + 2 cos(22pi t) )Which can be written in terms of exponentials as:( f(t) = frac{3}{2} e^{j4pi t} + frac{3}{2} e^{-j4pi t} + frac{7}{2} e^{j10pi t} + frac{7}{2} e^{-j10pi t} + frac{2}{2} e^{j22pi t} + frac{2}{2} e^{-j22pi t} )So, the Fourier coefficients ( c_n ) are:- For ( n = 20 ): ( c_{20} = frac{3}{2} )- For ( n = -20 ): ( c_{-20} = frac{3}{2} )- For ( n = 50 ): ( c_{50} = frac{7}{2} )- For ( n = -50 ): ( c_{-50} = frac{7}{2} )- For ( n = 110 ): ( c_{110} = frac{2}{2} = 1 )- For ( n = -110 ): ( c_{-110} = 1 )All other ( c_n ) are zero.Now, the power spectral density (PSD) is given by the magnitude squared of the Fourier transform. For a periodic signal, the PSD is a sum of delta functions at the harmonic frequencies, each with magnitude equal to the square of the Fourier coefficient.So, the PSD ( S(f) ) is:( S(f) = sum_{n=-infty}^{infty} |c_n|^2 delta(f - n f_0) )Where ( f_0 = 0.1 ) Hz.But in our case, the significant ( c_n ) are at ( n = pm20, pm50, pm110 ). So, the PSD will have impulses at ( f = pm2 ) Hz, ( pm5 ) Hz, and ( pm11 ) Hz.The magnitude at each of these frequencies is ( |c_n|^2 ). So:- At ( f = pm2 ) Hz: ( |c_{20}|^2 = (frac{3}{2})^2 = frac{9}{4} )- At ( f = pm5 ) Hz: ( |c_{50}|^2 = (frac{7}{2})^2 = frac{49}{4} )- At ( f = pm11 ) Hz: ( |c_{110}|^2 = (1)^2 = 1 )But wait, actually, in the PSD, each delta function's magnitude is ( |c_n|^2 ), but since the Fourier transform of a real signal is conjugate symmetric, the PSD is symmetric around zero. So, for each positive frequency, there is a corresponding negative frequency with the same magnitude.However, when calculating the total power, we can integrate the PSD over all frequencies, but for a periodic signal, the total power is the sum of the squares of the Fourier coefficients divided by the period (or something like that). Wait, let me recall.The average power of a periodic signal is given by:( P = frac{1}{T} int_{0}^{T} |f(t)|^2 dt )But since ( f(t) ) is a sum of sinusoids, we can use the orthogonality of sinusoids to compute the power.Each sinusoidal component contributes ( frac{A_n^2}{2} ) to the average power. So, the total average power is the sum of ( frac{A_n^2}{2} ) for all n.In our case, the amplitudes are 3, 7, and 2. So, the average power would be:( P = frac{3^2}{2} + frac{7^2}{2} + frac{2^2}{2} = frac{9}{2} + frac{49}{2} + frac{4}{2} = frac{62}{2} = 31 ) units.But wait, the problem asks for the total power over the 10-second period. Since average power is power per unit time, the total power would be average power multiplied by the period.So, total power ( P_{total} = P times T = 31 times 10 = 310 ) units.Alternatively, let's think about it another way. The total energy of the signal over one period is the integral of ( |f(t)|^2 ) over that period. Since the signal is periodic, the total energy over 10 seconds is just the energy over one period.Given that ( f(t) ) is a sum of sinusoids, the energy can be computed as the sum of the energies of each sinusoid. Each sinusoid ( A cos(omega t) ) has an average power of ( frac{A^2}{2} ), so over one period, the energy is ( frac{A^2}{2} times T ).So, for each component:- 3 cos(4πt): Energy = ( frac{3^2}{2} times 10 = frac{9}{2} times 10 = 45 )- 7 cos(10πt): Energy = ( frac{7^2}{2} times 10 = frac{49}{2} times 10 = 245 )- 2 cos(22πt): Energy = ( frac{2^2}{2} times 10 = frac{4}{2} times 10 = 20 )Adding them up: 45 + 245 + 20 = 310. So, total energy is 310 units, which is the same as total power over 10 seconds.Wait, but actually, power is energy per unit time. So, average power is 31 units per second, and over 10 seconds, the total energy is 310 units.But the problem says "total power of the signal over the 10-second period." Hmm, sometimes people use "power" to mean energy when referring to a specific time interval. So, in this context, I think they mean total energy, which is 310 units.Alternatively, if they strictly mean power, it would be 31 units per second, but since it's over 10 seconds, it's more likely they want the total energy, which is 310.But let me double-check the definitions. Power is energy per unit time, so the average power is 31 W (assuming units are in watts), and total energy is 310 J (joules). So, depending on the context, but since the problem mentions "total power," it's a bit ambiguous. However, in signal processing, when dealing with PSD, the total power is often referred to as the integral of PSD over all frequencies, which gives the average power. But in this case, since it's over a specific period, I think they want the total energy.But let's see, the PSD is the power per unit frequency. So, the total power is the integral of PSD over all frequencies, which for a periodic signal is the sum of the squared magnitudes of the Fourier coefficients. Wait, but in this case, the PSD is a sum of delta functions, so integrating over all frequencies would give the sum of the squared magnitudes, which is the average power.Wait, no, actually, the total power (average power) is the integral of the PSD over all frequencies. Since the PSD is ( S(f) ), the average power ( P ) is:( P = int_{-infty}^{infty} S(f) df )But for our case, ( S(f) ) is a sum of delta functions:( S(f) = frac{9}{4} [delta(f - 2) + delta(f + 2)] + frac{49}{4} [delta(f - 5) + delta(f + 5)] + 1 [delta(f - 11) + delta(f + 11)] )So, integrating this over all f:( P = frac{9}{4} times 2 + frac{49}{4} times 2 + 1 times 2 = frac{9}{2} + frac{49}{2} + 2 = frac{58}{2} + 2 = 29 + 2 = 31 )So, the average power is 31 units. Therefore, over 10 seconds, the total energy is ( 31 times 10 = 310 ) units.So, to summarize:- The function ( f(t) ) is a sum of cosines with the given frequencies and amplitudes.- The PSD consists of delta functions at ±2 Hz, ±5 Hz, and ±11 Hz with magnitudes ( frac{9}{4} ), ( frac{49}{4} ), and ( 1 ) respectively.- The total power (average power) is 31 units, so over 10 seconds, the total energy is 310 units.I think that covers both parts of the problem.</think>

🧠:<think>Okay, so I have this problem where I need to figure out the cost-effectiveness of ordering concrete mix from two different vendors, Vendor A and Vendor B. The store owner wants to know which vendor is cheaper for a 50-ton order and then how to split a 300-ton order between them to minimize costs while ensuring at least 100 tons from each. Hmm, let's break this down step by step.Starting with the first part: determining the cost per ton for an order of 50 tons from each vendor. I need to calculate the total cost and then divide by 50 to get the cost per ton.Vendor A:Vendor A sells at 50 per ton with a bulk discount. The discount is 5% for every 10 tons purchased, up to a maximum of 30%. So, for 50 tons, how much discount do they get?First, figure out how many 10-ton increments are in 50 tons. That's 50 / 10 = 5 increments. Each increment gives a 5% discount. So, 5 increments x 5% = 25%. But wait, the maximum discount is 30%, so 25% is within the limit. Therefore, the discount is 25%.So, the price per ton after discount is 50 - (25% of 50). Let me calculate that: 25% of 50 is 12.5, so 50 - 12.5 = 37.5 per ton.Wait, hold on. Is the discount applied per ton or on the total? Hmm, the problem says "bulk discount of 5% for every 10 tons purchased." So, it's a discount on the total price, not per ton. So, actually, maybe I should calculate the total cost first and then divide by 50 to get the cost per ton.Let me re-examine that. Vendor A's price is 50 per ton. For 50 tons, the total cost before discount is 50 x 50 = 2500.Now, the discount is 5% for every 10 tons. So for 50 tons, that's 5 increments of 10 tons. So 5 x 5% = 25% discount. So, the discount amount is 25% of 2500, which is 0.25 x 2500 = 625.Therefore, the total cost after discount is 2500 - 625 = 1875. Then, the cost per ton is 1875 / 50 = 37.5 per ton. Okay, so my initial calculation was correct.Vendor B:Vendor B sells at 55 per ton with a standard discount of 20% for purchases over 20 tons. Additionally, there's an extra 2% discount per 5 tons beyond the first 20 tons, up to a maximum discount of 40%.So, for 50 tons, we need to calculate the discount.First, the base discount is 20% for over 20 tons. So, that's 20%.Then, beyond 20 tons, we have 50 - 20 = 30 tons. For every 5 tons beyond 20, we get an additional 2% discount. So, how many 5-ton increments are in 30 tons? 30 / 5 = 6 increments. Each gives 2%, so 6 x 2% = 12%.Therefore, total discount is 20% + 12% = 32%. But wait, the maximum discount is 40%, so 32% is within the limit.So, the total discount is 32%. Therefore, the total cost before discount is 50 x 55 = 2750.Discount amount is 32% of 2750, which is 0.32 x 2750 = 880.Total cost after discount is 2750 - 880 = 1870.Therefore, the cost per ton is 1870 / 50 = 37.4 per ton.Wait, so Vendor A is 37.5 per ton, Vendor B is 37.4 per ton. So Vendor B is slightly cheaper by 0.10 per ton for a 50-ton order.But let me double-check my calculations because sometimes discounts can be applied differently.For Vendor A: 50 tons, 5 increments of 10 tons, 25% discount. 25% of 2500 is 625, so total cost 1875, which is 37.5 per ton. Correct.For Vendor B: 50 tons. First 20 tons get 20% discount, but actually, the 20% is a standard discount for purchases over 20 tons, so it's applied to the entire order? Or is it only on the amount over 20 tons?Wait, the problem says: "a standard discount of 20% for purchases over 20 tons, but with an additional 2% discount per 5 tons beyond the first 20 tons, up to a maximum discount of 40%."Hmm, so the 20% is a flat discount for purchasing over 20 tons. So, the entire order gets 20% discount. Then, beyond 20 tons, for each additional 5 tons, you get an extra 2% discount. So, it's 20% plus extra 2% per 5 tons beyond 20.So, for 50 tons, the first 20 tons get 20% discount, and the remaining 30 tons get 20% + (30/5)*2% = 20% + 12% = 32% discount on the entire order? Or is it 20% on the first 20 tons and 32% on the remaining 30 tons?Wait, the wording is a bit ambiguous. Let me read it again: "a standard discount of 20% for purchases over 20 tons, but with an additional 2% discount per 5 tons beyond the first 20 tons, up to a maximum discount of 40%."So, it's 20% for over 20 tons, and then additional 2% per 5 tons beyond 20. So, the total discount is 20% + (number of 5-ton increments beyond 20 tons) x 2%, up to 40%.So, for 50 tons, beyond 20 tons is 30 tons, which is 6 increments of 5 tons. So, 6 x 2% = 12%. Therefore, total discount is 20% + 12% = 32%.So, the total discount is 32%, applied to the entire order. So, total cost is 50 x 55 = 2750. 32% discount: 2750 x 0.32 = 880. So, total cost is 2750 - 880 = 1870. Therefore, cost per ton is 1870 / 50 = 37.4.Yes, so Vendor B is cheaper by 0.10 per ton for 50 tons.Wait, but let me think again. Is the discount applied per ton or on the total? The problem says "a standard discount of 20% for purchases over 20 tons." So, it's a discount on the total price. Similarly, the additional discounts are on the total price. So, yes, the 32% discount is applied to the entire 50 tons.So, Vendor A: 37.5 per ton.Vendor B: 37.4 per ton.So, Vendor B is slightly cheaper for 50 tons.Okay, moving on to the second part. The store projects selling 300 tons next month. They need to split the order between Vendor A and Vendor B, with at least 100 tons from each. So, the order split must be between 100 and 200 tons for each vendor.We need to find the optimal split that minimizes the total cost.To do this, I think we need to model the total cost as a function of the amount ordered from each vendor, considering their respective discount structures.Let me denote:Let x = tons ordered from Vendor A.Then, (300 - x) = tons ordered from Vendor B.Constraints: x >= 100, (300 - x) >= 100 => x <= 200.So, x is between 100 and 200.We need to find x in [100, 200] that minimizes the total cost.Total cost = Cost_A(x) + Cost_B(300 - x).We need expressions for Cost_A(x) and Cost_B(y), where y = 300 - x.First, let's define the cost functions for each vendor.Cost Function for Vendor A:Vendor A sells at 50 per ton with a bulk discount of 5% for every 10 tons, up to a maximum discount of 30%.So, for any quantity Q, the discount is min(5% * (Q / 10), 30%). So, discount rate = min(0.05 * (Q / 10), 0.30) = min(0.005 * Q, 0.30).Therefore, the cost is Q * 50 * (1 - discount rate).Similarly, for Vendor B:Vendor B sells at 55 per ton with a standard discount of 20% for purchases over 20 tons, plus an additional 2% per 5 tons beyond 20 tons, up to a maximum discount of 40%.So, for any quantity Q:If Q <= 20, discount is 0%.If Q > 20, discount is 20% + 2% * (Q - 20)/5, up to 40%.So, discount rate = min(0.20 + 0.02 * ((Q - 20)/5), 0.40).Simplify: discount rate = min(0.20 + 0.004 * (Q - 20), 0.40).So, for Q > 20, discount rate = 0.20 + 0.004*(Q - 20), but not exceeding 0.40.Therefore, the cost is Q * 55 * (1 - discount rate).Now, let's express these functions.Vendor A:For Q tons, discount rate d_A = min(0.005*Q, 0.30).So, Cost_A(Q) = Q * 50 * (1 - d_A).Vendor B:For Q tons, if Q <= 20, d_B = 0.If Q > 20, d_B = min(0.20 + 0.004*(Q - 20), 0.40).So, Cost_B(Q) = Q * 55 * (1 - d_B).Now, our total cost is Cost_A(x) + Cost_B(300 - x), where x is between 100 and 200.We need to find x that minimizes this total cost.To do this, we can model the cost functions and then find the minimum.But since the discount structures are piecewise linear, we need to consider different ranges where the discount rates change.First, let's analyze Vendor A's discount:For Vendor A, the discount increases by 5% every 10 tons, up to 30%.So, for Q:- 0-10 tons: 0% discount- 10-20 tons: 5%- 20-30 tons: 10%- 30-40 tons: 15%- 40-50 tons: 20%- 50-60 tons: 25%- 60-70 tons: 30%- Beyond 70 tons: 30% (max)But in our case, x can be up to 200 tons. So, for Vendor A, the discount rate for x tons is:d_A = min(0.005*x, 0.30).So, for x <= 60 tons, d_A increases by 5% every 10 tons. For x > 60, d_A is 30%.But in our problem, x is between 100 and 200 tons. So, for Vendor A, when x is 100 tons, d_A = min(0.005*100, 0.30) = min(0.5, 0.30) = 0.30. So, for x >= 60, d_A is 30%.Therefore, for x in [100, 200], Vendor A's discount is 30%.Therefore, Cost_A(x) = x * 50 * (1 - 0.30) = x * 50 * 0.70 = 35x.So, Vendor A's cost is linear in x for x >= 60 tons.Now, Vendor B's discount:For Vendor B, the discount is 20% for Q > 20, plus 2% per 5 tons beyond 20, up to 40%.So, for Q = 300 - x, which is between 100 and 200 tons (since x is between 100 and 200).So, Q is between 100 and 200.So, for Q > 20, which it is, the discount is 20% + 2% per 5 tons beyond 20.So, discount rate d_B = 0.20 + 0.02*(Q - 20)/5.Simplify: d_B = 0.20 + 0.004*(Q - 20).But since Q is between 100 and 200, let's compute d_B:d_B = 0.20 + 0.004*(Q - 20) = 0.20 + 0.004Q - 0.08 = 0.12 + 0.004Q.But we also have a maximum discount of 40%, so d_B cannot exceed 0.40.So, let's see when d_B reaches 0.40:0.12 + 0.004Q = 0.400.004Q = 0.28Q = 0.28 / 0.004 = 70 tons.So, for Q >= 70 tons, d_B would be 0.40.But in our case, Q is between 100 and 200, so d_B is 0.40.Therefore, for Q >= 70, d_B = 40%.Therefore, for Vendor B, when Q is between 100 and 200, d_B is 40%.Therefore, Cost_B(Q) = Q * 55 * (1 - 0.40) = Q * 55 * 0.60 = 33Q.So, for Q >= 70, Vendor B's cost is linear with a rate of 33 per ton.Wait, but let me confirm:If Q is 100 tons, then d_B = 0.20 + 0.004*(100 - 20) = 0.20 + 0.004*80 = 0.20 + 0.32 = 0.52. But the maximum discount is 40%, so d_B is capped at 0.40.Therefore, for Q >= 70, d_B = 0.40.So, yes, for Q >= 70, Vendor B's cost is 33 per ton.Therefore, for both vendors, when ordering more than 60 tons (for Vendor A) and more than 70 tons (for Vendor B), their cost per ton becomes fixed at 35 and 33 respectively.But in our case, x is between 100 and 200 for Vendor A, and Q = 300 - x is between 100 and 200 for Vendor B. So, both are in their maximum discount ranges.Therefore, the total cost is:Total Cost = 35x + 33*(300 - x) = 35x + 9900 - 33x = 2x + 9900.Wait, that can't be right. Because 35x + 33*(300 - x) = 35x + 9900 - 33x = 2x + 9900.So, the total cost is 2x + 9900.But this is a linear function in x, with a positive coefficient (2). Therefore, to minimize the total cost, we need to minimize x, since increasing x increases the total cost.But x must be at least 100 tons. So, the minimum total cost occurs when x = 100 tons.Therefore, ordering 100 tons from Vendor A and 200 tons from Vendor B.Let me verify this.If x = 100:Cost_A = 35 * 100 = 3500.Cost_B = 33 * 200 = 6600.Total Cost = 3500 + 6600 = 10,100.If x = 200:Cost_A = 35 * 200 = 7000.Cost_B = 33 * 100 = 3300.Total Cost = 7000 + 3300 = 10,300.So, indeed, the total cost increases as x increases. Therefore, to minimize the total cost, we should order as much as possible from the cheaper vendor, which is Vendor B, and the minimum required from Vendor A.But wait, Vendor B is cheaper per ton when ordering large quantities, but is this always the case?Wait, in the first part, for 50 tons, Vendor B was cheaper. For larger quantities, since both have maximum discounts, Vendor B is cheaper per ton (33 vs 35). So, yes, to minimize cost, order as much as possible from Vendor B, which is 200 tons, and 100 tons from Vendor A.Therefore, the optimal order split is 100 tons from Vendor A and 200 tons from Vendor B, with a total cost of 10,100.But let me double-check the cost functions to ensure I didn't make a mistake.For Vendor A, when x >= 60, cost per ton is 35.For Vendor B, when Q >= 70, cost per ton is 33.Yes, so 35 vs 33, so Vendor B is cheaper. Therefore, to minimize cost, maximize the order from Vendor B, subject to the constraint of at least 100 tons from each.Therefore, 100 from A, 200 from B.Total cost: 100*35 + 200*33 = 3500 + 6600 = 10,100.Yes, that seems correct.But just to be thorough, let's check if ordering more from Vendor B beyond 200 tons is possible, but since the total is 300, and we need at least 100 from A, 200 is the maximum from B.Alternatively, if we order 150 from A and 150 from B, what would the cost be?But wait, for Vendor A, 150 tons is within the 30% discount, so cost is 35*150 = 5250.For Vendor B, 150 tons is within the 40% discount, so cost is 33*150 = 4950.Total cost: 5250 + 4950 = 10,200, which is higher than 10,100.Similarly, ordering 120 from A and 180 from B:Cost_A = 35*120 = 4200.Cost_B = 33*180 = 5940.Total: 4200 + 5940 = 10,140, which is still higher than 10,100.Therefore, the minimal total cost is indeed when ordering 100 from A and 200 from B.So, summarizing:1. For 50 tons, Vendor B is cheaper at 37.4 per ton vs Vendor A's 37.5.2. For 300 tons, order 100 tons from Vendor A and 200 tons from Vendor B, with a total cost of 10,100.</think>

🧠:<think>Alright, so I have this problem about curating a photography exhibition. Let me try to figure it out step by step. First, the gallery is a rectangle with two walls of 30 meters and two walls of 20 meters. So, total wall space would be 2*(30+20) = 100 meters. Got that.Problem 1: There are 24 photographs, each 1.5 meters wide. They need to be placed with a minimum of 0.5 meters between them. I need to find the total length required and check if they can all fit.Okay, so each photo is 1.5m, and between each, there's 0.5m. Since there are 24 photos, how many spaces are there between them? Well, if you have n items, there are n-1 spaces between them. So, 24 photos mean 23 spaces.So, total space required would be (24*1.5) + (23*0.5). Let me calculate that.24*1.5 is 36 meters. 23*0.5 is 11.5 meters. So, total is 36 + 11.5 = 47.5 meters.Now, the gallery has 100 meters of wall space. 47.5 is less than 100, so yes, they can all be displayed. But wait, the problem says "if the photographs can be hung on all four walls." So, does that mean they can spread them out on all walls? Hmm, but the total required is 47.5, which is less than 100, so yes, they can fit.Wait, but maybe I need to check if each wall can accommodate some photos without exceeding their lengths. Let me think.The walls are 30, 30, 20, 20 meters. So, if we spread the photos across all four walls, each wall can have some number of photos.But actually, the total required space is 47.5 meters, which is less than 100. So, regardless of how we distribute them, as long as each wall isn't exceeded, it's fine. But maybe we need to check if the arrangement on each wall is possible.Wait, maybe not. Since the total required is 47.5, and the total available is 100, so yes, they can fit. So, the answer is 47.5 meters, and yes, they can be displayed.Problem 2: Now, the editor wants to form groups of 3 consecutive photos, each group needing an additional narrative panel of 2 meters. So, for every 3 photos, there's a 2m panel.First, how many groups are there? Since there are 24 photos, and each group is 3 photos, that's 24/3 = 8 groups.Each group requires 2m, so total panels space is 8*2 = 16 meters.But wait, each group is 3 photos, so for each group, we have 3 photos and 1 panel. So, the total space per group would be the space for 3 photos plus the panel.Wait, but the photos themselves take up space, and the panels also take up space. So, for each group, we need to calculate the space for the 3 photos with spacing, plus the panel.Wait, the original spacing is between photos, but now, between groups, do we need additional spacing? Hmm, the problem says "for every 3 consecutive photographs, there needs to be an additional narrative display panel of 2 meters width." So, I think that means after every 3 photos, there's a panel. So, the sequence would be: photo, space, photo, space, photo, panel, photo, space, etc.Wait, but the original spacing is 0.5m between each photo. So, for 3 photos, it's 3*1.5 + 2*0.5 = 4.5 + 1 = 5.5 meters. Then, after that, a panel of 2 meters. So, each group (3 photos + panel) takes 5.5 + 2 = 7.5 meters.But wait, if we have 8 groups, each taking 7.5 meters, that would be 8*7.5 = 60 meters. But wait, is that correct?Wait, no, because the panels are in between the groups. So, if we have 8 groups, there would be 7 panels between them? Wait, no, the problem says "for every 3 consecutive photographs, there needs to be an additional narrative display panel." So, each group of 3 photos has its own panel. So, 8 groups, 8 panels.But in terms of placement, after each group of 3 photos, there is a panel. So, the total arrangement would be: group1 (3 photos + panel), group2 (3 photos + panel), etc. So, each group is 3 photos, 2m panel, and between groups, do we need spacing? The problem doesn't specify, so maybe the panels are placed right after the group, without extra spacing.But wait, the original spacing is between photos, which is 0.5m. So, within each group, the 3 photos have 0.5m between them, and then after the third photo, a 2m panel.So, for each group: 3 photos take 3*1.5 + 2*0.5 = 4.5 + 1 = 5.5m, plus 2m panel, total 7.5m per group.Since there are 8 groups, total space is 8*7.5 = 60 meters.But wait, let me think again. If each group is 3 photos with spacing, plus a panel, and then the next group starts after that. So, the total would be 8 groups * (3*1.5 + 2*0.5 + 2) = 8*(4.5 + 1 + 2) = 8*7.5 = 60 meters.But wait, the total wall space is 100 meters. So, 60 meters is less than 100, so they can fit. But wait, is that the correct way to calculate?Alternatively, maybe the panels are placed between the groups, so if there are 8 groups, there are 7 panels between them. But the problem says "for every 3 consecutive photographs, there needs to be an additional narrative display panel." So, each group of 3 photos has its own panel, regardless of other groups. So, 8 groups, 8 panels.But in terms of placement, if you have 8 groups, each with 3 photos and a panel, the total would be 8*(3*1.5 + 2*0.5 + 2) = 8*(4.5 + 1 + 2) = 8*7.5 = 60 meters.But wait, another way: the total photos are 24, each 1.5m, so 24*1.5=36m. The spaces between photos: 23*0.5=11.5m. The panels: 8*2=16m. So, total is 36 + 11.5 + 16 = 63.5 meters.Wait, that's different from 60. So which is correct?Hmm, the confusion is whether the panels are placed within the sequence of photos, adding to the total length, or if they are placed in addition to the photos.Wait, the problem says "the narrative panels also need to be placed within the gallery along with the photographs." So, they are placed along the walls, so they take up additional space.So, the total space is photos + spaces between photos + panels.So, photos: 24*1.5=36m.Spaces between photos: 23*0.5=11.5m.Panels: 8*2=16m.Total: 36 + 11.5 + 16 = 63.5 meters.Since the gallery has 100 meters, 63.5 < 100, so yes, they can fit.But wait, another thought: when forming groups, the panels are placed after every 3 photos, so the spacing between the panels and the next group? Or is the panel part of the group's space?Wait, the problem says "for every 3 consecutive photographs, there needs to be an additional narrative display panel of 2 meters width." So, each group of 3 photos must have a panel. So, the panel is in addition to the photos and their spacing.So, the total space is photos + spaces between photos + panels.So, 24 photos: 36m.23 spaces: 11.5m.8 panels: 16m.Total: 63.5m.Yes, that makes sense.So, the total wall space required is 63.5 meters, which is less than 100, so they can fit.Wait, but in the first problem, the total was 47.5m, and now it's 63.5m. So, the addition of panels increases the required space.So, the answer for problem 2 is 63.5 meters, and yes, they can fit.But let me double-check.Alternatively, if we consider that each group of 3 photos plus a panel takes up 3*1.5 + 2*0.5 + 2 = 4.5 + 1 + 2 = 7.5m. With 8 groups, that's 8*7.5=60m. But then, between the groups, do we need to add spacing? The problem doesn't specify, so maybe not. So, total is 60m.But then, why is the other method giving 63.5m?I think the confusion is whether the panels are placed within the sequence, meaning that after each group of 3 photos, a panel is placed, and then the next group starts. So, the spacing between the panel and the next group is not specified, so perhaps it's zero. So, the total would be 8*(3*1.5 + 2*0.5 + 2) = 60m.But the other approach counts all photos, all spaces between photos, and all panels, which gives 63.5m.Which is correct?I think the correct approach is to consider that the panels are placed after each group, so the total space is the sum of all photos, all spaces between photos, and all panels.Because the panels are additional elements that take up space, so they need to be added to the total.So, 24 photos: 36m.23 spaces: 11.5m.8 panels: 16m.Total: 63.5m.Yes, that makes sense.So, the total wall space required is 63.5 meters, which is less than 100, so they can fit.Therefore, the answers are:1. Total space: 47.5 meters, yes.2. Total space: 63.5 meters, yes.</think>

🧠:<think>Alright, so I have these two probability problems to solve. Let me take them one at a time. Starting with the first problem: Vinyl Vic has 5,000 vinyl records. 20% are rare live recordings, and the rest are studio albums. Among the rare live recordings, 70% are from the blues-rock genre of the 70s. I need to find the probability that if he randomly selects 3 records without replacement, all three are rare live recordings from the blues-rock genre.Okay, so let's break this down. First, how many rare live recordings does he have? 20% of 5,000 is 0.2 * 5000 = 1000 records. Out of these 1000, 70% are blues-rock from the 70s. So that's 0.7 * 1000 = 700 records.So, there are 700 favorable records out of 5000 total. He's selecting 3 without replacement, so this is a hypergeometric probability problem, right? Because we're dealing with successes and failures without replacement.The formula for hypergeometric probability is:P = (C(K, k) * C(N - K, n - k)) / C(N, n)Where:- N is the total population size (5000)- K is the number of success states in the population (700)- n is the number of draws (3)- k is the number of observed successes (3)So plugging in the numbers:P = (C(700, 3) * C(5000 - 700, 3 - 3)) / C(5000, 3)Simplify that:C(700, 3) is the number of ways to choose 3 blues-rock live records from 700.C(4300, 0) is 1, since there's only one way to choose nothing.So P = C(700, 3) / C(5000, 3)Calculating combinations:C(n, k) = n! / (k! * (n - k)!)So C(700, 3) = 700! / (3! * 697!) = (700 * 699 * 698) / (3 * 2 * 1) = let's compute that.700 * 699 = 489,300489,300 * 698 = Hmm, that's a big number. Let me see:First, 489,300 * 700 = 342,510,000But since it's 698, which is 700 - 2, so 489,300 * 698 = 489,300*(700 - 2) = 489,300*700 - 489,300*2 = 342,510,000 - 978,600 = 341,531,400Then divide by 6 (since 3! is 6):341,531,400 / 6 = 56,921,900Wait, let me double-check that multiplication:Wait, 700 * 699 = 489,300, correct. Then 489,300 * 698: Let's compute 489,300 * 698.Compute 489,300 * 700 = 342,510,000Subtract 489,300 * 2 = 978,600So 342,510,000 - 978,600 = 341,531,400, yes. Then divide by 6: 341,531,400 / 6 = 56,921,900.So C(700, 3) is 56,921,900.Now, C(5000, 3): Let's compute that.C(5000, 3) = 5000! / (3! * 4997!) = (5000 * 4999 * 4998) / 6Compute numerator: 5000 * 4999 = 24,995,00024,995,000 * 4998: Hmm, that's a huge number. Let me compute step by step.First, 24,995,000 * 5000 = 124,975,000,000But since it's 4998, which is 5000 - 2, so 24,995,000 * 4998 = 24,995,000*(5000 - 2) = 24,995,000*5000 - 24,995,000*2Compute 24,995,000*5000: 24,995,000 * 5,000 = 124,975,000,000Compute 24,995,000*2 = 49,990,000Subtract: 124,975,000,000 - 49,990,000 = 124,925,010,000Wait, no: 124,975,000,000 - 49,990,000 is 124,925,010,000? Wait, 124,975,000,000 minus 49,990,000 is 124,925,010,000? Wait, 124,975,000,000 minus 50,000,000 would be 124,925,000,000, but since it's 49,990,000, it's 10,000 more. So 124,925,000,000 + 10,000 = 124,925,010,000. Yes, that's correct.So numerator is 124,925,010,000.Divide by 6: 124,925,010,000 / 6 = 20,820,835,000.Wait, let me check:6 * 20,820,835,000 = 124,925,010,000, yes.So C(5000, 3) is 20,820,835,000.Therefore, the probability P is 56,921,900 / 20,820,835,000.Let me compute that.First, simplify numerator and denominator:Divide numerator and denominator by 100: 569,219 / 208,208,350.Hmm, let's see if we can simplify further. Let's see if 569,219 divides into 208,208,350.Compute 208,208,350 / 569,219 ≈ Let's see:569,219 * 365 = ?Well, 569,219 * 300 = 170,765,700569,219 * 60 = 34,153,140569,219 * 5 = 2,846,095Add them together: 170,765,700 + 34,153,140 = 204,918,840 + 2,846,095 = 207,764,935So 569,219 * 365 ≈ 207,764,935But our denominator is 208,208,350, which is a bit more.Compute 208,208,350 - 207,764,935 = 443,415So 569,219 * 365 + 443,415 = 208,208,350So 443,415 / 569,219 ≈ 0.779So total is approximately 365.779So 569,219 / 208,208,350 ≈ 1 / 365.779 ≈ 0.002734So approximately 0.002734, which is about 0.2734%.Wait, but let me compute it more accurately.Alternatively, compute 56,921,900 / 20,820,835,000.Divide numerator and denominator by 100: 569,219 / 208,208,350.Compute 569,219 ÷ 208,208,350.Let me write it as 569219 / 208208350.Divide numerator and denominator by GCD(569219, 208208350). Let's see if 569219 divides into 208208350.Compute 208208350 ÷ 569219 ≈ 365.779, as before.So it's approximately 1 / 365.779 ≈ 0.002734.So approximately 0.2734%.Alternatively, using decimal division:569219 ÷ 208208350.Let me write it as 569219 ÷ 208208350 ≈ 0.002734.So the probability is approximately 0.2734%.Wait, that seems low, but considering there are only 700 favorable out of 5000, and selecting 3, it's not too surprising.Alternatively, maybe I can compute it as:Probability of first record being blues-rock live: 700/5000Then, without replacement, second is 699/4999Third is 698/4998Multiply them together:(700/5000) * (699/4999) * (698/4998)Compute each fraction:700/5000 = 0.14699/4999 ≈ 0.139827965698/4998 ≈ 0.139679719Multiply them together:0.14 * 0.139827965 ≈ 0.019575915Then 0.019575915 * 0.139679719 ≈ 0.002734So same result, about 0.002734, or 0.2734%.So that seems consistent.So the probability is approximately 0.2734%, or 0.002734.So that's the first problem.Now, moving on to the second problem: The collection follows a Poisson distribution for the number of albums added per year with a mean (λ) of 50. What is the probability that Vinyl Vic adds at least 60 albums in a given year?So, Poisson distribution: P(X = k) = (λ^k * e^{-λ}) / k!We need P(X ≥ 60) = 1 - P(X ≤ 59)But computing this directly would require summing from k=0 to k=59, which is tedious.Alternatively, since λ is 50, which is moderately large, we can use the normal approximation to the Poisson distribution.The Poisson distribution with λ=50 can be approximated by a normal distribution with μ=λ=50 and σ=√λ≈7.0711.So, we can compute P(X ≥ 60) ≈ P(Z ≥ (60 - 50)/7.0711) = P(Z ≥ 1.4142)Looking up the standard normal distribution table, P(Z ≥ 1.41) is approximately 0.0793, and P(Z ≥ 1.42) is approximately 0.0778.Since 1.4142 is approximately 1.41, so maybe around 0.0793.But let me compute it more accurately.Alternatively, use the continuity correction. Since we're approximating a discrete distribution with a continuous one, we can adjust by 0.5.So, P(X ≥ 60) ≈ P(X ≥ 59.5) in the normal distribution.So, compute Z = (59.5 - 50)/7.0711 ≈ 9.5 / 7.0711 ≈ 1.343So, P(Z ≥ 1.343). Looking up 1.34 in the Z-table: 0.0901, and 1.35 is 0.0885.Since 1.343 is closer to 1.34, maybe around 0.0901 - (0.0901 - 0.0885)*(0.3/1) ≈ 0.0901 - 0.0016*0.3 ≈ 0.0901 - 0.00048 ≈ 0.0896.Wait, actually, the Z-score is 1.343, which is 1.34 + 0.003.The difference between 1.34 and 1.35 is 0.0885 to 0.0901, which is a difference of 0.0016 over 0.01 in Z.So, per 0.001 increase in Z, the probability decreases by 0.0016/0.01 = 0.16 per 0.001.So, for 0.003, it would decrease by 0.16 * 0.003 = 0.00048.So, starting at Z=1.34: 0.0901Subtract 0.00048: 0.0901 - 0.00048 ≈ 0.08962.So, approximately 0.0896, or 8.96%.Alternatively, using a calculator or precise Z-table, but I think 8.96% is a reasonable approximation.Alternatively, using the Poisson formula directly, but that would require summing from k=60 to infinity, which is impractical by hand.Alternatively, using the complement: 1 - P(X ≤ 59). But again, summing 60 terms is tedious.Alternatively, use the normal approximation without continuity correction: Z=(60 - 50)/7.0711≈1.4142, which is about 1.41.Looking up Z=1.41: 0.0793.But with continuity correction, it's about 0.0896.Which one is better? The continuity correction usually gives a better approximation, so I think 8.96% is more accurate.Alternatively, using the Poisson cumulative distribution function, perhaps using software, but since I don't have that, I'll go with the normal approximation with continuity correction.So, approximately 8.96% chance.Wait, but let me think again. The exact value can be computed using the Poisson CDF, but without a calculator, it's tough. Alternatively, use the fact that for Poisson, the probability of X ≥ μ + kσ can be approximated, but I think the normal approximation is acceptable here.So, I think the answer is approximately 8.96%, or 0.0896.But let me check another way.Alternatively, using the Poisson PMF and summing from 60 to infinity, but that's not feasible manually.Alternatively, use the recursive formula for Poisson probabilities, but that's also time-consuming.Alternatively, use the fact that for Poisson, the probability of X ≥ k is equal to 1 - the sum from i=0 to k-1 of (λ^i e^{-λ}) / i!But again, without computation, it's hard.Alternatively, use the fact that for λ=50, the distribution is roughly bell-shaped, so the probability of being above 60 is about the same as being more than 1.41σ above the mean, which is about 7.93%, but with continuity correction, it's about 8.96%.Alternatively, perhaps use the Poisson cumulative distribution function approximation.Wait, another method: using the relationship between Poisson and chi-squared distributions.But that might be overcomplicating.Alternatively, use the fact that for Poisson, the probability can be approximated using the normal distribution with continuity correction.So, I think the answer is approximately 8.96%.But to get a better approximation, perhaps use the fact that the normal approximation with continuity correction is better.So, I think I'll go with approximately 8.96%, or 0.0896.Alternatively, let's compute it more precisely.Using the normal approximation with continuity correction:Z = (59.5 - 50)/√50 ≈ 9.5 / 7.0711 ≈ 1.343Looking up Z=1.343 in standard normal table.Using a precise Z-table, Z=1.34 is 0.0901, Z=1.35 is 0.0885.The difference between 1.34 and 1.35 is 0.0016 in probability.Since 1.343 is 0.003 above 1.34, so the probability decreases by (0.003/0.01)*0.0016 = 0.00048.So, 0.0901 - 0.00048 ≈ 0.08962.So, approximately 0.0896, or 8.96%.Alternatively, using linear interpolation:At Z=1.34, P=0.0901At Z=1.35, P=0.0885The difference in Z is 0.01, and the difference in P is -0.0016.So, per 0.001 increase in Z, P decreases by 0.0016/0.01 = 0.16 per 0.001.So, for Z=1.343, which is 0.003 above 1.34, the decrease is 0.003 * 0.16 = 0.00048.Thus, P=0.0901 - 0.00048=0.08962.So, approximately 0.0896, or 8.96%.Alternatively, using a calculator, the exact value using normal CDF is about 0.0896.So, I think that's the answer.Alternatively, if I use the Poisson formula, the exact probability is:P(X ≥ 60) = 1 - P(X ≤ 59)But without computation, I can't get the exact value, but I know that the normal approximation with continuity correction is better, so 8.96% is a reasonable estimate.So, to sum up:1. The probability is approximately 0.2734%, or 0.002734.2. The probability is approximately 8.96%, or 0.0896.Wait, but let me check the first problem again.Wait, I think I made a mistake in the first problem. Because when I computed C(700,3)/C(5000,3), I got approximately 0.002734, but let me confirm.Alternatively, using the product of probabilities:(700/5000) * (699/4999) * (698/4998)Compute each term:700/5000 = 0.14699/4999 ≈ 0.139827965698/4998 ≈ 0.139679719Multiply them:0.14 * 0.139827965 ≈ 0.019575915Then 0.019575915 * 0.139679719 ≈ 0.002734Yes, same result.Alternatively, using combinations:C(700,3) = 700*699*698/(3*2*1) = 56,921,900C(5000,3) = 5000*4999*4998/(3*2*1) = 20,820,835,000So, 56,921,900 / 20,820,835,000 ≈ 0.002734Yes, correct.So, the first answer is approximately 0.2734%, or 0.002734.The second answer is approximately 8.96%, or 0.0896.I think that's it.</think>