💡Alright, so I have this problem about maximizing monthly profit for a product. Let me try to understand it step by step. First, the cost price is 40 yuan per unit, and the selling price is initially 50 yuan per unit. They can sell 210 units per month at this price. Now, if they increase the selling price by 1 yuan, the number of units sold decreases by 10 per month. Also, the selling price can't go above 65 yuan. Okay, so they want me to find the relationship between the monthly profit (y) and the increase in selling price (x), where x is a positive integer. Then, they want to know at what price they should sell to maximize the profit and what that maximum profit is.Let me start by figuring out the profit function. Profit is usually calculated as (Selling Price - Cost Price) multiplied by the number of units sold. So, if the selling price increases by x yuan, the new selling price becomes (50 + x) yuan. The cost price remains 40 yuan, so the profit per unit is (50 + x - 40) = (10 + x) yuan.Now, the number of units sold decreases by 10 for each 1 yuan increase. So, if the price increases by x yuan, the number of units sold becomes (210 - 10x). Therefore, the total monthly profit y should be the product of the profit per unit and the number of units sold. So, y = (10 + x) * (210 - 10x). Let me write that out:y = (10 + x)(210 - 10x)Hmm, let me expand this to make it a quadratic function. First, multiply 10 by 210: 10*210 = 2100Then, 10 by -10x: 10*(-10x) = -100xThen, x by 210: x*210 = 210xAnd x by -10x: x*(-10x) = -10x²So, putting it all together:y = 2100 - 100x + 210x - 10x²Combine like terms:-100x + 210x = 110xSo, the equation becomes:y = -10x² + 110x + 2100Okay, that looks right. Now, I need to make sure about the range of x. The selling price can't exceed 65 yuan. The initial price is 50, so the maximum increase x can be is 65 - 50 = 15 yuan. Also, x has to be a positive integer, so x can be from 1 to 15.Wait, but in the problem statement, it says "x is a positive integer," so x starts at 1. But sometimes, in these problems, x can be zero as well. Let me check. If x is zero, that means no increase, so the selling price remains 50 yuan. The number of units sold would still be 210. So, y would be (10 + 0)*(210 - 0) = 10*210 = 2100 yuan. But since x is a positive integer, I think x starts at 1. So, the range of x is 1 ≤ x ≤ 15.Alright, so part (1) is done. The functional relationship is y = -10x² + 110x + 2100, with x ranging from 1 to 15 inclusive.Now, part (2) asks at what price each product should be sold to maximize the monthly profit and what that maximum profit is.Since we have a quadratic function in terms of x, and the coefficient of x² is negative (-10), the parabola opens downward, meaning the vertex is the maximum point.The general form of a quadratic function is y = ax² + bx + c, and the vertex occurs at x = -b/(2a). In our case, a = -10 and b = 110. So, plugging into the formula:x = -110 / (2*(-10)) = -110 / (-20) = 5.5Hmm, x = 5.5. But x has to be an integer between 1 and 15. So, we need to check the profit at x = 5 and x = 6, since 5.5 is halfway between 5 and 6.Let me calculate y at x = 5:y = -10*(5)² + 110*(5) + 2100= -10*25 + 550 + 2100= -250 + 550 + 2100= 300 + 2100= 2400 yuanNow, at x = 6:y = -10*(6)² + 110*(6) + 2100= -10*36 + 660 + 2100= -360 + 660 + 2100= 300 + 2100= 2400 yuanInteresting, both x = 5 and x = 6 give the same profit of 2400 yuan. So, the maximum profit is 2400 yuan, and it occurs at both x = 5 and x = 6.Therefore, the selling price should be either 50 + 5 = 55 yuan or 50 + 6 = 56 yuan per unit. Both prices will yield the maximum monthly profit of 2400 yuan.Let me just double-check my calculations to make sure I didn't make any errors.For x = 5:- Selling price = 55 yuan- Units sold = 210 - 10*5 = 210 - 50 = 160 units- Profit per unit = 55 - 40 = 15 yuan- Total profit = 160 * 15 = 2400 yuanFor x = 6:- Selling price = 56 yuan- Units sold = 210 - 10*6 = 210 - 60 = 150 units- Profit per unit = 56 - 40 = 16 yuan- Total profit = 150 * 16 = 2400 yuanYep, both give 2400 yuan. So, that seems correct.Just to ensure completeness, let me check x = 4 and x = 7 to see if the profit is indeed lower, as expected.For x = 4:- Selling price = 54 yuan- Units sold = 210 - 40 = 170 units- Profit per unit = 14 yuan- Total profit = 170 * 14 = 2380 yuanFor x = 7:- Selling price = 57 yuan- Units sold = 210 - 70 = 140 units- Profit per unit = 17 yuan- Total profit = 140 * 17 = 2380 yuanYes, both x = 4 and x = 7 give a lower profit of 2380 yuan, confirming that x = 5 and x = 6 are indeed the points of maximum profit.So, summarizing:1. The profit function is y = -10x² + 110x + 2100, with x ranging from 1 to 15.2. The maximum profit of 2400 yuan occurs when the selling price is either 55 yuan or 56 yuan per unit.I think that covers everything the problem is asking for.

💡First, I need to determine how many cell phones each machine produces per minute. The problem states that some machines make 10 cell phones each minute. Next, it mentions that it would take 10 machines to make 50 cell phones per minute. To find out how many cell phones each machine makes, I can divide the total number of cell phones by the number of machines: 50 cell phones divided by 10 machines equals 5 cell phones per machine per minute.However, there seems to be a discrepancy because the initial statement says that some machines make 10 cell phones each minute, while the calculation suggests each machine makes 5 cell phones per minute. Assuming the correct rate is 5 cell phones per machine per minute, then 10 machines would indeed produce 50 cell phones per minute. Therefore, the machines make 50 cell phones each minute.

💡First, I need to determine the number of men among the 300 passengers. Since 55% of the passengers are men, I calculate 55% of 300.Next, I need to find out how many of these men are in the luxury coach. Given that 15% of the men are in the luxury coach, I calculate 15% of the number of men.Finally, I perform the calculations to find the exact number of men in the luxury coach.

💡First, I need to determine the radius of the circle with a diameter of 15 inches. Since the radius is half of the diameter, I divide 15 by 2 to get 7.5 inches.Next, I calculate the area of both circles using the formula ( A = pi r^2 ). For the circle with a radius of 25 inches, the area is ( pi times 25^2 = 625pi ) square inches. For the circle with a radius of 7.5 inches, the area is ( pi times 7.5^2 = 56.25pi ) square inches.Finally, I find the difference between the two areas by subtracting the smaller area from the larger one: ( 625pi - 56.25pi = 568.75pi ) square inches.

💡First, I need to calculate the total feet climbed for each of the four staircases.For the first staircase, there are 24 steps, each 0.6 feet high. Multiplying the number of steps by the height per step gives the total height climbed for this staircase.Next, the second staircase has three times as many steps as the first one, which means there are 72 steps. Each step in this staircase is 0.4 feet high. Again, multiplying the number of steps by the height per step will give the total height climbed here.Moving on to the third staircase, it has 20 fewer steps than the second staircase, resulting in 52 steps. Each step here is 0.5 feet high. Multiplying the number of steps by the height per step will provide the total height climbed for this staircase.Finally, the fourth staircase has half as many steps as the first one, which is 12 steps. Each step in this staircase is double the height of the first staircase, making each step 1.2 feet high. Multiplying the number of steps by the height per step will give the total height climbed for this staircase.After calculating the total feet climbed for each staircase, I will add them all together to find the overall total feet climbed by John.

💡Okay, so I have this problem where a student did a physics experiment and got some data points. The data is in a table with x and y values. The x values are 1, 2, 3, 4, 5, and the corresponding y values are 5.380, 11.232, 20.184, 34.356, and 53.482. I need to figure out which function model fits this data best among the four options given: A, B, C, and D.First, let me understand what each option represents.Option A is a linear function: y = ax + b, where a is greater than 1. So, this is a straight line with a slope greater than 1. If I plot y against x, it should be a straight line that increases as x increases.Option B is a square root function: y = a√x + b, with a > 1. This means that as x increases, y increases, but the rate of increase slows down because the square root of x grows slower as x gets larger.Option C is a reciprocal function: y = a/x + b, with a > 1. This function decreases as x increases because as x gets larger, a/x gets smaller. However, in our data, y increases as x increases, so this might not be the right fit.Option D is a quadratic function: y = ax² + b, with a > 1. This is a parabola that opens upwards since a is positive. The y-values will increase as x increases, and the rate of increase will accelerate, which might match the data if the y-values are increasing more rapidly as x increases.Now, let's look at the data points:When x = 1, y = 5.380x = 2, y = 11.232x = 3, y = 20.184x = 4, y = 34.356x = 5, y = 53.482I notice that as x increases, y increases quite significantly. Let's see if this increase is linear, quadratic, or something else.First, let's check if it's linear. For a linear function, the difference between consecutive y-values should be roughly constant.Calculating the differences:11.232 - 5.380 = 5.85220.184 - 11.232 = 8.95234.356 - 20.184 = 14.17253.482 - 34.356 = 19.126These differences are 5.852, 8.952, 14.172, and 19.126. They are increasing, which suggests that the rate of change is not constant. Therefore, a linear model (Option A) might not be the best fit.Next, let's consider the square root function (Option B). If y = a√x + b, then as x increases, y increases, but the rate of increase should slow down. Let's see:Calculate the differences in y-values again:From x=1 to x=2: 5.852From x=2 to x=3: 8.952From x=3 to x=4: 14.172From x=4 to x=5: 19.126These differences are increasing, meaning the rate of increase is getting faster, not slower. This contradicts the behavior of a square root function, which should have decreasing differences. So, Option B might not be suitable.Option C is a reciprocal function, y = a/x + b. As x increases, y should decrease, but in our data, y increases as x increases. This is the opposite of what we observe, so Option C is likely not the right choice.That leaves us with Option D, the quadratic function y = ax² + b. Quadratic functions have the property that the rate of change increases as x increases, which matches what we see in the data—the differences between consecutive y-values are getting larger as x increases.To confirm, let's see if the second differences are constant, which is a characteristic of quadratic functions.First differences (as calculated before):5.852, 8.952, 14.172, 19.126Now, calculate the second differences:8.952 - 5.852 = 3.114.172 - 8.952 = 5.2219.126 - 14.172 = 4.954These second differences are approximately 3.1, 5.22, and 4.954. They are not exactly constant, but they are somewhat close, especially considering that the data might not be perfectly quadratic due to experimental errors or other factors.Given that the second differences are somewhat consistent and the first differences are increasing, it suggests that a quadratic model (Option D) is the most suitable among the given options.Therefore, after analyzing the data and comparing it to the characteristics of each function model, I conclude that Option D is the best fit.