Compute the maximum product for two positive numbers x and y with the property that the sum of the first plus five times the second is 5000.
1) Indicate the objective equation
2) Indicate the constraint equation
So the objective = f(x)=xy and the constraint = x+5y=5000
Now number 3 says rewrite the objective function in terms of the variable x only. Would it be y=x?
4) For which value of x does the objective function attain its maximum? Do I have to take the derivative?
5) What is the corresponding value of y
6) What is the vale of the maximum product of the two numbers?
Nevermind, my question was answered already. Thank you.
1) The objective equation is f(x) = xy, representing the maximum product of two positive numbers x and y.
2) The constraint equation is x + 5y = 5000, indicating that the sum of the first number x and five times the second number y is 5000.
3) To rewrite the objective function in terms of the variable x only, we can substitute the value of y from the constraint equation into the objective equation. From the constraint equation, we have y = (5000 - x)/5. Substituting this into the objective equation, we get f(x) = x * (5000 - x)/5.
4) To find the value of x that maximizes the objective function, we can take the derivative of the rewritten objective function with respect to x, set it equal to 0, and solve for x. By finding the critical point of the function, we can determine where the maximum occurs.
5) Once we find the value of x that maximizes the objective function, we can substitute it back into the constraint equation (x + 5y = 5000) to solve for y. This will give us the corresponding value of y.
6) Finally, to find the maximum product of the two numbers, we substitute the values of x and y into the objective function f(x) = xy. This will yield the maximum product.
1) The objective equation is f(x)=xy, where x and y are positive numbers representing the two numbers.
2) The constraint equation is x + 5y = 5000, representing the property that the sum of the first number plus five times the second number is equal to 5000.
3) To rewrite the objective function in terms of the variable x only, you can substitute the constraint equation into the objective equation. From the constraint equation, we can rearrange it to get y = (5000 - x) / 5. Substituting this expression for y in the objective equation, we get f(x) = x * ((5000 - x) / 5).
4) To find the value of x that maximizes the objective function, you can take the derivative of the objective function with respect to x and set it equal to zero. Taking the derivative of f(x) = x * ((5000 - x) / 5) with respect to x, we get f'(x) = (5000 - 6x) / 5. Setting f'(x) equal to zero and solving for x, we get (5000 - 6x) / 5 = 0. Solving this equation, we find x = 5000 / 6.
5) To find the corresponding value of y, we can substitute the value of x obtained in the previous step into the constraint equation. Substituting x = 5000 / 6 into x + 5y = 5000, we can solve for y and find y = 5000 / 6.
6) The maximum product of the two numbers can be found by substituting the values of x and y obtained in the previous steps into the objective equation. Substituting x = 5000 / 6 and y = 5000 / 6 into f(x) = xy, we get f(x) = (5000 / 6) * (5000 / 6), which simplifies to a value you can calculate.