Suppose that 430 ft of fencing is used to enclose a corral in the shape of a rectangle with a semicircle whose diameter is a side of the rectangle. (In the figure below, the blue outline represents the fencing.) Find the dimensions of the corral with maximum area.

x=?
y=?

i tried couple of ways but got the wrong answer, i know i have to take the derivative and find the minimum value for x but i don't know how to set it up.

To find the dimensions of the corral with maximum area, we can set up an equation using the given information and then apply calculus to find the optimal values. Let's break down the problem step by step:

1. Let's start by drawing a diagram to visualize the corral. We have a rectangle with sides x and y, and a semicircle with a diameter of x.

2. The perimeter of the corral is given as 430 ft. We can write the equation for the perimeter as:
Perimeter = 2x + y + πx/2

3. Since the perimeter is given as 430 ft, we can write the equation:
2x + y + πx/2 = 430

4. We need to express the area of the corral in terms of x and y. The area of the rectangle is A_rect = xy, and the area of the semicircle is half the area of a full circle, which is A_circle = (πx²)/8.

5. The total area of the corral is the sum of the rectangle and the semicircle:
A_total = A_rect + A_circle
A_total = xy + (πx²)/8

6. Now, we want to find the dimensions (x and y) that maximize the area (A_total). To do this, we need to find the critical points by taking the derivative of A_total with respect to x.

7. Take the derivative:
dA_total/dx = d(xy + (πx²)/8)/dx
dA_total/dx = y + (πx)/4

8. Now, set the derivative equal to zero to find the critical points:
y + (πx)/4 = 0

9. Rearrange the equation to solve for y:
y = -(πx)/4

10. Substitute this value of y back into the original equation for the perimeter:
2x + (-(πx)/4) + πx/2 = 430
Simplify the equation:
2x - (πx)/4 + πx/2 = 430
Multiply the equation by 4 to get rid of the fractional coefficient:
8x - πx + 2πx = 1720
Simplify further:
(8 - π + 2π)x = 1720
(8 + π)x = 1720
x = 1720 / (8 + π)

11. Now that we have found the value of x, substitute it back into the equation for y that we derived earlier:
y = -(π(1720 / (8 + π)))/4

12. Calculate the actual values of x and y using a calculator or another mathematical tool.

So, the dimensions of the corral with maximum area are x = 1720 / (8 + π) and y = -(π(1720 / (8 + π)))/4.