🧠:<think>To find the intercept form of the equation (2x - 3y - 4 = 0), I need to determine the x-intercept and y-intercept of the line.First, I'll find the x-intercept by setting (y = 0) in the equation:[2x - 3(0) - 4 = 0 implies 2x - 4 = 0 implies 2x = 4 implies x = 2]So, the x-intercept is at ((2, 0)).Next, I'll find the y-intercept by setting (x = 0) in the equation:[2(0) - 3y - 4 = 0 implies -3y - 4 = 0 implies -3y = 4 implies y = -frac{4}{3}]So, the y-intercept is at ((0, -frac{4}{3})).The intercept form of a line is given by:[frac{x}{a} + frac{y}{b} = 1]where (a) is the x-intercept and (b) is the y-intercept.Substituting (a = 2) and (b = -frac{4}{3}) into the intercept form:[frac{x}{2} + frac{y}{-frac{4}{3}} = 1]This matches option D.</think>

🧠:<think>Okay, so I have this probability problem here, and I need to estimate from below the probability that the deviation in the frequency of the expected result from the average probability does not exceed 0.04 in absolute value. Let me try to break this down step by step.First, let me understand the setup. There are 800 independent trials. In 200 of these trials, the probability of the expected outcome is 0.5. In 400 trials, the probability is 0.4, and in the remaining 200 trials, the probability is 0.3. So, to recap: 200 trials with p=0.5, 400 trials with p=0.4, and 200 trials with p=0.3.I need to find the average probability across all these trials. That should be straightforward. The average probability, let's denote it as (bar{p}), can be calculated by taking the weighted average of all the individual probabilities.So, (bar{p} = frac{(200 times 0.5) + (400 times 0.4) + (200 times 0.3)}{800}).Let me compute each part:- (200 times 0.5 = 100)- (400 times 0.4 = 160)- (200 times 0.3 = 60)Adding these up: (100 + 160 + 60 = 320).Now, divide by the total number of trials: (320 / 800 = 0.4).So, the average probability (bar{p}) is 0.4. Got that.Now, the problem is about estimating the probability that the deviation in the frequency of the expected result from this average probability does not exceed 0.04. In other words, we want the probability that the observed frequency (m/n) is within 0.04 of 0.4.Mathematically, this can be written as:[ Pleft( left| frac{m}{n} - 0.4 right| leq 0.04 right) ]I remember that for such problems, especially when dealing with large numbers of trials, the Law of Large Numbers or Chebyshev's inequality might be useful. But since the trials have different probabilities, it's not a simple binomial distribution. Maybe I need to use Poisson's inequality or something similar.Wait, Poisson's theorem? I think that's related to the convergence of the frequency to the probability. Let me recall.Poisson's theorem states that for a sequence of independent trials with different probabilities (p_i), the probability that the frequency (m/n) deviates from the average probability (bar{p}) by more than (epsilon) is bounded by:[ Pleft( left| frac{m}{n} - bar{p} right| geq epsilon right) leq frac{sum_{i=1}^{n} p_i (1 - p_i)}{n^2 epsilon^2} ]So, to find the lower bound of the probability that the deviation does not exceed (epsilon), we can rearrange this inequality:[ Pleft( left| frac{m}{n} - bar{p} right| < epsilon right) geq 1 - frac{sum_{i=1}^{n} p_i (1 - p_i)}{n^2 epsilon^2} ]Alright, so in our case, (epsilon = 0.04), (n = 800), and (bar{p} = 0.4). I need to compute the sum (sum_{i=1}^{n} p_i (1 - p_i)).Given that the trials are divided into three groups with different probabilities, I can compute this sum by considering each group separately.So, for the first group: 200 trials with (p = 0.5). The term (p(1 - p)) for each trial is (0.5 times 0.5 = 0.25). So, for 200 trials, the total contribution is (200 times 0.25 = 50).For the second group: 400 trials with (p = 0.4). The term (p(1 - p)) is (0.4 times 0.6 = 0.24). So, for 400 trials, the total contribution is (400 times 0.24 = 96).For the third group: 200 trials with (p = 0.3). The term (p(1 - p)) is (0.3 times 0.7 = 0.21). So, for 200 trials, the total contribution is (200 times 0.21 = 42).Adding these contributions together: (50 + 96 + 42 = 188).So, the numerator in our inequality is 188. The denominator is (n^2 epsilon^2), which is (800^2 times 0.04^2).Calculating (800^2): (800 times 800 = 640,000).Calculating (0.04^2): (0.04 times 0.04 = 0.0016).Multiplying these together: (640,000 times 0.0016 = 1,024).So, the denominator is 1,024.Now, plugging these into our inequality:[ Pleft( left| frac{m}{n} - 0.4 right| < 0.04 right) geq 1 - frac{188}{1,024} ]Calculating (frac{188}{1,024}): Let's see, 188 divided by 1,024.Well, 1,024 divided by 16 is 64, so 188 divided by 1,024 is approximately 0.18359375.So, (1 - 0.18359375 = 0.81640625).Rounding this to three decimal places, it's approximately 0.816.Therefore, the lower bound for the probability that the deviation in the frequency of the expected result from the average probability does not exceed 0.04 is approximately 0.816.Wait, let me double-check my calculations to make sure I didn't make any mistakes.First, the average probability: 200*0.5 + 400*0.4 + 200*0.3 = 100 + 160 + 60 = 320. 320/800 = 0.4. That seems correct.Sum of p_i(1 - p_i): 200*(0.5*0.5) + 400*(0.4*0.6) + 200*(0.3*0.7) = 50 + 96 + 42 = 188. That looks right.Denominator: 800^2 * 0.04^2 = 640,000 * 0.0016 = 1,024. Correct.188 / 1,024 = 0.18359375. So, 1 - 0.18359375 = 0.81640625, which is approximately 0.816. Rounded to three decimal places, that's 0.816.So, yes, that seems correct. Therefore, the lower bound is approximately 0.816.I think that's the answer. It's a bit involved, but breaking it down step by step helped me understand each part of the problem.</think>

🧠:<think>Okay, so I have this problem where there's a function ( g(x) ) that satisfies the equation ( g(xy) = frac{g(x)}{y} ) for all positive real numbers ( x ) and ( y ). I'm told that ( g(24) = 12 ), and I need to find ( g(36) ). Hmm, functional equations can sometimes be tricky, but let me see if I can figure this out step by step.First, let me try to understand what the equation ( g(xy) = frac{g(x)}{y} ) is telling me. It seems like if I multiply the input ( x ) by some positive real number ( y ), the output of the function gets divided by ( y ). So, scaling the input by ( y ) inversely scales the output by ( y ). That's interesting. Maybe this suggests that ( g(x) ) has some kind of inverse relationship with ( x )?Let me think about what kind of function would satisfy this property. If I consider ( g(x) ) as being proportional to ( frac{1}{x} ), that might work because if ( g(x) = frac{k}{x} ) for some constant ( k ), then ( g(xy) = frac{k}{xy} ), which is equal to ( frac{g(x)}{y} ) because ( frac{g(x)}{y} = frac{k}{x y} ). So, yes, that seems to fit.So, if ( g(x) = frac{k}{x} ), then we can use the given information ( g(24) = 12 ) to find the constant ( k ). Let's do that:( g(24) = frac{k}{24} = 12 )To solve for ( k ), I can multiply both sides by 24:( k = 12 times 24 )Calculating that, 12 times 24 is 288. So, ( k = 288 ). Therefore, the function ( g(x) ) is:( g(x) = frac{288}{x} )Now, to find ( g(36) ), I can substitute 36 into the function:( g(36) = frac{288}{36} )Calculating that, 288 divided by 36 is 8. So, ( g(36) = 8 ).Wait, let me double-check that to make sure I didn't make a mistake. If ( g(x) = frac{288}{x} ), then ( g(24) = frac{288}{24} = 12 ), which matches the given information. Then, ( g(36) = frac{288}{36} = 8 ). Yep, that seems correct.Alternatively, maybe I can solve this without assuming the form of ( g(x) ). Let me try that approach too to confirm.Given the functional equation ( g(xy) = frac{g(x)}{y} ), I can choose specific values for ( x ) and ( y ) to find ( g(36) ). Since I know ( g(24) = 12 ), maybe I can express 36 as 24 multiplied by some ( y ), and then use the functional equation.Let me set ( x = 24 ) and choose ( y ) such that ( xy = 36 ). So, ( 24 times y = 36 ). Solving for ( y ), I get ( y = frac{36}{24} = frac{3}{2} ).Now, using the functional equation:( g(24 times frac{3}{2}) = frac{g(24)}{frac{3}{2}} )Simplifying the left side, ( 24 times frac{3}{2} = 36 ), so:( g(36) = frac{g(24)}{frac{3}{2}} )We know ( g(24) = 12 ), so:( g(36) = frac{12}{frac{3}{2}} )Dividing by a fraction is the same as multiplying by its reciprocal, so:( g(36) = 12 times frac{2}{3} = 8 )Okay, that's another way to get the same answer, which is reassuring. So, whether I find the general form of ( g(x) ) or use the functional equation directly, I end up with ( g(36) = 8 ).Just to make sure I haven't overlooked anything, let me consider if there are other possible functions that satisfy the given equation. Suppose there's another function ( h(x) ) that also satisfies ( h(xy) = frac{h(x)}{y} ). If I set ( x = 1 ), then ( h(y) = frac{h(1)}{y} ), which suggests that ( h(x) = frac{h(1)}{x} ). So, indeed, the function must be of the form ( frac{k}{x} ) where ( k ) is a constant. Therefore, my initial assumption about the form of ( g(x) ) was correct.Another way to think about this is to consider the functional equation as a definition of how the function scales. If I increase the input by a factor of ( y ), the output decreases by the same factor ( y ). This is characteristic of an inverse proportionality, which again points to ( g(x) ) being proportional to ( frac{1}{x} ).So, putting it all together, since ( g(x) = frac{k}{x} ) and ( g(24) = 12 ), solving for ( k ) gives ( k = 288 ). Therefore, ( g(36) = frac{288}{36} = 8 ).I think I've covered all the angles here. I've derived the function using the given condition, checked it with specific values, and even considered alternative approaches to confirm the result. I don't see any mistakes in my reasoning, so I feel confident that ( g(36) = 8 ) is the correct answer.Final AnswerThe value of ( g(36) ) is boxed{8}.</think>

🧠:<think>First, I need to determine the probability of successfully placing a lasso around a cow's neck in a single throw. Let's denote this probability as ( p ).The probability of not being able to place a lasso in a single throw is ( 1 - p ).The probability of not being able to place a lasso in 3 consecutive throws is ( (1 - p)^3 ).The probability of being able to place a lasso at least once in 3 attempts is the complement of not being able to do so in any of the 3 attempts, which is ( 1 - (1 - p)^3 ).Given that this probability is 0.875, I can set up the equation:[ 1 - (1 - p)^3 = 0.875 ]Solving for ( p ):[ (1 - p)^3 = 1 - 0.875 ][ (1 - p)^3 = 0.125 ]Taking the cube root of both sides:[ 1 - p = sqrt[3]{0.125} ][ 1 - p = 0.5 ]Therefore, ( p = 1 - 0.5 = 0.5 ).The average number of throws it takes for the rancher to place a lasso around a cow's neck is the reciprocal of the probability of success in a single throw:[ text{Average number of throws} = frac{1}{p} ][ text{Average number of throws} = frac{1}{0.5} ][ text{Average number of throws} = 2 ]So, on average, it takes the rancher boxed{2} throws to place a lasso around a cow's neck.</think>