Question

Suppose that 435ft of fencing are used to enclose a corral in the shape of a rectangle with a semicircle whose diameter is a side of a rectangle as the following figure: Find the dimensions of the corral with maximum area.x=___ft.y=___ft.

Answer 1

The dimensions of the corral with maximum area are approximately
`\( x = 96.96 \) ft and \( y = 24.54 \)` ft, achieved by using 435 ft of fencing to enclose a rectangle with a semicircular end.

To find the dimensions of the corral with the maximum area, we can set up an equation based on the given information.

Let \( x \) be the width of the rectangle, and \( y \) be the length of the rectangle. The total amount of fencing used is the sum of the perimeter of the rectangle and the circumference of the semicircle.

The perimeter of the rectangle is
`\( 2x + 2y \),` and the circumference of the semicircle is
`\( \pi * \text{radius} \),` where the radius is
`\( (x)/(2) \)`(half of the diameter, which is the width of the rectangle).

So, the total amount of fencing is given by the equation:

`\[ 2x + 2y + \pi * (x)/(2) = 435 \]`

Now, we can solve this equation for one of the variables and substitute it into the area formula to maximize the area.

`\[ 2x + 2y + \pi * (x)/(2) = 435 \]`

Combine like terms:

`\[ 2x + 2y + (\pi x)/(2) = 435 \]`

Multiply through by 2 to eliminate the fraction:

`\[ 4x + 4y + \pi x = 870 \]`

Now, solve for \( y \) in terms of \( x \):

\[ 4y = 870 - 4x - \pi x \]

`\[ y = (870)/(4) - x - (\pi x)/(4) \]`

`\[ y = 217.5 - x - 0.25 \pi x \]`

Now, we can express the area (\( A \)) in terms of \( x \) by multiplying \( x \) and \( y \):

\[ A(x) = x \times \left(217.5 - x - 0.25 \pi x\right) \]

`\[ A(x) = 217.5x - x^2 - 0.25 \pi x^2 \]`

To find the maximum area, take the derivative of \( A(x) \) with respect to \( x \) and set it equal to zero:

`\[ (dA)/(dx) = 217.5 - 2x - 0.5 \pi x = 0 \]`

Solving for \( x \):

`\[ 2x + 0.5 \pi x = 217.5 \]`

`\[ x(2 + 0.5 \pi) = 217.5 \]`

`\[ x = (217.5)/(2 + 0.5 \pi) \]`

Now, substitute this value of \( x \) back into the equation for \( y \) to find the corresponding \( y \) value.

`\[ y = 217.5 - x - 0.25 \pi x \]`