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 the maximum area, we can set up an equation based on the given information.
Let's assume the length of the rectangle is x. Therefore, the width of the rectangle is also x.
Next, let's consider the semicircle. The diameter of the semicircle is equal to the width of the rectangle, which is x. The radius of the semicircle is half the diameter, so it is x/2.
The circumference of the semicircle is π times the radius. Therefore, the length of the straight part of the fence at the top and bottom of the rectangle is equal to the circumference of the semicircle, which is π(x/2) = (πx)/2.
The length of the straight part of the fence on the sides of the rectangle is equal to 2 times the width of the rectangle, which is 2x.
Adding up all the lengths of the fences, we get:
Length of the fence = 2x + (πx)/2 + 2x
From the information given in the problem, we know that the total length of the fence is 430 ft. So we can set up the equation:
2x + (πx)/2 + 2x = 430
Now, we can solve this equation to find the value of x.
First, let's simplify the equation:
4x + (πx)/2 = 430
Multiplying through by 2 to eliminate the fraction:
8x + πx = 860
Combining like terms:
(8 + π)x = 860
Dividing both sides by (8 + π):
x = 860 / (8 + π)
Now that we have found the value of x, we can substitute it back into the equation for the length of the fence to find the value of y (the width of the rectangle):
y = x
Therefore, the dimensions of the corral with the maximum area are:
x = 860 / (8 + π)
y = 860 / (8 + π)