🧠:<think>First, I need to calculate the annual maintenance cost for the data protection system over five years, considering that the cost increases by 5% each year. The initial maintenance cost is 15,000.For Year 1, the maintenance cost is 15,000. For Year 2, it increases by 5%, so I multiply 15,000 by 1.05 to get 15,750. I'll apply the same 5% increase for Years 3, 4, and 5, resulting in costs of 16,537.50, 17,364.38, and 18,232.59 respectively.Next, I'll sum up all the maintenance costs over the five years and add the initial investment of 15,000 to find the total cost.For the second part, I need to determine the expected quarterly revenue after two years, assuming an 8% growth rate per quarter. The initial quarterly revenue is 500,000. I'll calculate the revenue for each quarter over the two years by applying the 8% growth rate sequentially.After calculating each quarter's revenue, I'll identify the revenue at the end of the eighth quarter, which will be the expected quarterly revenue after two years.</think>

🧠:<think>Okay, so I have this problem where a sports journalist is analyzing an athlete's performance over a decade. There are two parts: one about the number of games played and another about the scoring average per game. Let me try to figure out each part step by step.Starting with the first part: the Games Played Model. The function given is ( G(t) = a cdot e^{kt} ). Here, ( t = 0 ) is the starting year, ( a ) is the initial number of games, and ( k ) is a growth constant. We know two things: in the first year (( t = 0 )), the athlete played 50 games, and in the fifth year (( t = 5 )), they played 100 games. I need to find ( a ) and ( k ).Alright, let's plug in the first piece of information. When ( t = 0 ), ( G(0) = 50 ). So substituting into the equation:( G(0) = a cdot e^{k cdot 0} )Since ( e^{0} = 1 ), this simplifies to:( 50 = a cdot 1 )So, ( a = 50 ). That was straightforward.Now, moving on to find ( k ). We know that in the fifth year (( t = 5 )), the athlete played 100 games. So, substituting into the equation:( G(5) = 50 cdot e^{k cdot 5} = 100 )Let me write that equation:( 50e^{5k} = 100 )To solve for ( k ), I can divide both sides by 50:( e^{5k} = 2 )Now, to solve for ( 5k ), I'll take the natural logarithm of both sides:( ln(e^{5k}) = ln(2) )Simplifying the left side:( 5k = ln(2) )So, ( k = frac{ln(2)}{5} )Calculating that, ( ln(2) ) is approximately 0.6931, so:( k approx frac{0.6931}{5} approx 0.1386 )So, ( k ) is approximately 0.1386 per year. Let me just double-check my steps. Plugging ( t = 5 ) into the equation with ( a = 50 ) and ( k approx 0.1386 ):( G(5) = 50e^{0.1386 cdot 5} )Calculating the exponent: 0.1386 * 5 = 0.693So, ( e^{0.693} ) is approximately 2, which gives ( 50 * 2 = 100 ). Perfect, that checks out.Alright, so part 1 is done. ( a = 50 ) and ( k approx 0.1386 ).Moving on to part 2: the Scoring Average Model. The function is given as ( S(t) = mt + b ), a linear function. Here, ( t ) is the number of years since the start of the decade. We know two things: in the first year (( t = 0 )), the scoring average was 20 points per game, and in the tenth year (( t = 10 )), it was 30 points per game. We need to find ( m ) and ( b ), and then predict the scoring average in the 15th year.Starting with the first data point: when ( t = 0 ), ( S(0) = 20 ). Plugging into the equation:( 20 = m cdot 0 + b )So, ( b = 20 ). Simple enough.Now, using the second data point: when ( t = 10 ), ( S(10) = 30 ). Plugging into the equation:( 30 = m cdot 10 + 20 )Subtract 20 from both sides:( 10 = 10m )So, ( m = 1 ).Therefore, the equation is ( S(t) = t + 20 ).Wait, hold on. Let me verify. If ( t = 10 ), then ( S(10) = 10 + 20 = 30 ). Correct. And ( t = 0 ) gives 20. Perfect.Now, to predict the scoring average in the 15th year, we plug ( t = 15 ) into the equation:( S(15) = 1 cdot 15 + 20 = 15 + 20 = 35 )So, the predicted scoring average is 35 points per game.But wait, let me think about this. The model is linear, so it's assuming a constant increase each year. From year 0 to year 10, the scoring average increased by 10 points over 10 years, which is a rate of 1 point per year. So, in the 15th year, it would be 5 years beyond the 10th year, so 5 more points, making it 35. That makes sense.Is there anything else I need to consider? Hmm, the problem says to assume the same trend continues, so a linear model is appropriate here. I don't think there's any need for a different model unless specified, so I think 35 is the correct prediction.Let me recap:1. For the games played model, we had an exponential function. We found ( a = 50 ) and ( k approx 0.1386 ).2. For the scoring average, it was a linear model. We found ( m = 1 ) and ( b = 20 ), predicting 35 points in the 15th year.I think that's all. I don't see any mistakes in my calculations, and the logic seems sound. So, I'm confident with these answers.Final Answer1. ( a = boxed{50} ) and ( k = boxed{frac{ln 2}{5}} )2. ( m = boxed{1} ), ( b = boxed{20} ), and the predicted scoring average in the 15th year is ( boxed{35} ) points per game.</think>

🧠:<think>First, I need to determine the time spent on each activity based on the given percentages. The father has 16 hours of productive time each day.1. Time Allocation Modeling: - Cooking and Cleaning: 40% of 16 hours is calculated by multiplying 0.40 by 16. - Homework and School Activities: 25% of 16 hours is calculated by multiplying 0.25 by 16. - Household Maintenance: 20% of 16 hours is calculated by multiplying 0.20 by 16. - Personal Relaxation: The remaining percentage is 15%, so I multiply 0.15 by 16 to find this time.2. Optimizing Task Efficiency: - The father decides to reduce cooking and cleaning by 10%. I calculate 10% of the original cooking and cleaning time and subtract it from the original amount. - This reduction allows an increase in household maintenance by the same number of hours. I add the reduced amount to the original household maintenance time. - The percentages for homework and school activities, as well as personal relaxation, remain unchanged since there is no reallocation affecting them. - Finally, I calculate the new percentage of time dedicated to household maintenance by dividing the new household maintenance time by the total productive time and multiplying by 100.By following these steps, I can provide the exact hours spent on each activity before and after the reallocation, as well as the new percentage for household maintenance.</think>

🧠:<think>Okay, so I have this problem about a small business owner who sells handmade items with gold leaf designs. The problem has two parts, and I need to solve both. Let me take it step by step.Starting with part 1: The owner wants to maximize profit. They have a function for the selling price, p(g) = 50 + 30√g, where g is the amount of gold leaf in square centimeters. The cost of the gold leaf is given by c(g) = 10g + 5. I need to find the optimal g that maximizes profit.First, I remember that profit is calculated as total revenue minus total cost. So, profit π = revenue - cost. But in this case, since we're dealing with individual items, I think the profit per item would be selling price minus cost. Wait, actually, the functions given are for each item, right? So, for each item, the selling price is p(g), and the cost is c(g). So, profit per item would be p(g) - c(g). Therefore, to maximize profit, we need to maximize p(g) - c(g) with respect to g.Let me write that down:Profit per item, π(g) = p(g) - c(g) = [50 + 30√g] - [10g + 5]Simplify that:π(g) = 50 + 30√g - 10g - 5 = 45 + 30√g - 10gSo, π(g) = -10g + 30√g + 45Now, to find the maximum profit, I need to find the value of g that maximizes π(g). Since this is a calculus problem, I should take the derivative of π with respect to g, set it equal to zero, and solve for g.Let's compute the derivative π'(g):π'(g) = d/dg [ -10g + 30√g + 45 ]The derivative of -10g is -10.The derivative of 30√g is 30*(1/(2√g)) = 15/√g.The derivative of 45 is 0.So, π'(g) = -10 + 15/√gSet π'(g) = 0 to find critical points:-10 + 15/√g = 0Let me solve for g:15/√g = 10Divide both sides by 5:3/√g = 2Wait, 15 divided by 10 is 1.5, which is 3/2. So, 15/√g = 10 => 15/10 = √g => 3/2 = √gWait, no, that's not correct. Let me re-express:15/√g = 10Multiply both sides by √g:15 = 10√gDivide both sides by 10:15/10 = √g => 3/2 = √gThen, square both sides:(3/2)^2 = g => 9/4 = g => g = 2.25So, the critical point is at g = 2.25 square centimeters.Now, I need to check if this critical point is a maximum. Since the profit function is a function of g, and we're dealing with a single variable, we can use the second derivative test or analyze the behavior of the first derivative around the critical point.Let me compute the second derivative π''(g):π'(g) = -10 + 15/√gSo, π''(g) = derivative of π'(g) = derivative of (-10) + derivative of (15/√g)Derivative of -10 is 0.Derivative of 15/√g is 15 * (-1/2) * g^(-3/2) = -15/(2g^(3/2))So, π''(g) = -15/(2g^(3/2))At g = 2.25, which is positive, π''(g) is negative because of the negative sign. Therefore, the function is concave down at this point, which means it's a local maximum. So, g = 2.25 is indeed the value that maximizes profit.Therefore, the optimal amount of gold leaf is 2.25 square centimeters.Wait, just to make sure I didn't make a mistake in the derivative:π(g) = -10g + 30√g + 45π'(g) = -10 + (30)*(1/(2√g)) = -10 + 15/√gYes, that's correct. Then setting equal to zero:15/√g = 10 => √g = 15/10 = 3/2 => g = (3/2)^2 = 9/4 = 2.25Yes, that seems right.Okay, so part 1 is done. The optimal g is 2.25.Moving on to part 2: The demand D for the handmade items is inversely proportional to the price p, so D(p) = k/p, where k is a constant. We need to find k when the owner sells 20 items at a price of 80. Then, calculate how many items would be sold at the profit-maximizing price.First, find k. Given that D(p) = k/p, and when p = 80, D = 20.So, plug in:20 = k / 80Solve for k:k = 20 * 80 = 1600So, k is 1600. Therefore, the demand function is D(p) = 1600 / p.Now, we need to find how many items are sold at the profit-maximizing price. From part 1, we found that the optimal g is 2.25. So, first, let's find the price p at g = 2.25.Given p(g) = 50 + 30√gSo, plug in g = 2.25:p = 50 + 30√(2.25)Compute √(2.25): that's 1.5, because 1.5^2 = 2.25.So, p = 50 + 30*1.5 = 50 + 45 = 95So, the profit-maximizing price is 95.Now, plug this into the demand function D(p) = 1600 / p.So, D(95) = 1600 / 95Compute that:1600 divided by 95. Let me compute that.95 * 16 = 15201600 - 1520 = 80So, 1600 / 95 = 16 + 80/95Simplify 80/95: divide numerator and denominator by 5: 16/19So, D(95) = 16 + 16/19 ≈ 16.842But since we can't sell a fraction of an item, maybe we need to round it? Or perhaps it's acceptable as a decimal.But the question says "how many items the owner would sell", so probably just compute it as a number, maybe as a fraction.So, 1600 / 95 simplifies:Divide numerator and denominator by 5: 320 / 19320 divided by 19: 19*16=304, 320-304=16, so 16 and 16/19.So, 16 16/19 items.But in reality, you can't sell a fraction of an item, so maybe the owner would sell 16 items. But since the question doesn't specify, perhaps we can leave it as 1600/95 or 320/19.Alternatively, maybe it's better to write it as a decimal: approximately 16.842, which is roughly 16.84. But since it's about items sold, maybe we need to round it to the nearest whole number, which would be 17 items.But let me check: 95 * 16 = 1520, 95*17=1615. Since 1600 is between 1520 and 1615, so 1600 is 1600 - 1520 = 80 above 1520, which is 80/95 ≈ 0.842 of the way to 17. So, 16.842.But since the demand function is D(p) = 1600 / p, which is a continuous function, but in reality, the number of items sold must be an integer. So, depending on how the owner sets the price, they might adjust the price slightly to sell a whole number of items. But since the question is asking for how many items would be sold at the profit-maximizing price, which is 95, we can just compute D(95) = 1600 / 95, which is approximately 16.84, but since you can't sell a fraction, perhaps the owner would sell 16 or 17 items.But maybe in the context of the problem, it's acceptable to have a fractional number of items sold, as a measure of demand, even if in reality you can't sell a fraction. So, maybe we can just leave it as 1600/95, which is 320/19, or approximately 16.84.But let me check the exact value:1600 divided by 95:95 goes into 160 once (95), remainder 65.Bring down 0: 650.95 goes into 650 six times (95*6=570), remainder 80.Bring down 0: 800.95 goes into 800 eight times (95*8=760), remainder 40.Bring down 0: 400.95 goes into 400 four times (95*4=380), remainder 20.Bring down 0: 200.95 goes into 200 twice (95*2=190), remainder 10.Bring down 0: 100.95 goes into 100 once (95), remainder 5.Bring down 0: 50.95 goes into 50 zero times, so we have 0.842105...So, 1600/95 ≈ 16.842105...So, approximately 16.84. So, if we have to give an exact value, it's 320/19, which is approximately 16.84.But the question says "how many items the owner would sell at the profit-maximizing price." So, maybe they expect the exact fractional form, which is 320/19, or they might accept the decimal.Alternatively, maybe I made a mistake in interpreting the demand function. Let me double-check.Demand D(p) is inversely proportional to p, so D(p) = k/p.Given that when p = 80, D = 20, so k = 20*80 = 1600. So, D(p) = 1600/p.At p = 95, D = 1600 / 95 = 320/19 ≈16.84.So, that's correct.Alternatively, maybe the owner can adjust the price slightly to sell a whole number, but since the question is about the profit-maximizing price, which is 95, the corresponding demand is 1600/95, which is approximately 16.84. So, unless the owner is willing to change the price, they would have to sell either 16 or 17 items. But since the question is about how many would be sold at the profit-maximizing price, it's just the value from the demand function, which is 1600/95.So, I think the answer is 320/19 or approximately 16.84. But since the question didn't specify, I think it's safer to write it as a fraction, 320/19.Alternatively, maybe I should write it as 1600/95, but simplifying 1600/95: divide numerator and denominator by 5: 320/19, which is the simplified form.So, the number of items sold is 320/19, which is approximately 16.84.But since the problem is about a small business owner, maybe they can only sell whole items, so perhaps the answer is 16 or 17. But the question doesn't specify rounding, so I think the exact value is 320/19.So, to recap:1. The optimal g is 2.25 square centimeters.2. The constant k is 1600, and the number of items sold at the profit-maximizing price is 320/19, approximately 16.84.Wait, just to make sure I didn't make a mistake in calculating p at g=2.25:p(g) = 50 + 30√gg=2.25, so √2.25=1.5So, p=50 + 30*1.5=50+45=95. Yes, that's correct.And D(p)=1600/p, so D(95)=1600/95=320/19≈16.84. Correct.So, I think that's all. I don't see any mistakes in my reasoning.</think>