A rancher wants to build a rectangular pen with an area of 180 . Let W be the width of the pen and L be the length of the pen.

a) Find an equation for the perimeter P in terms of W and L .

b) Use the given area to write an equation that relates W and L .

c) Find the pen dimensions that require the minimum amount of fencing.
Width =
Length =

I need the answer urgent! Thank you. Please provide an explanation on how you did it!

P=2(W+L)

A=WL, so L = 180/W

so,

P = 2(W+180/W)

This is pre-cal, so you don't have calculus at your disposal. But, you do know that a square has maximum area for a given perimeter. So,
W = L = √180 = 6√5

a) To find the perimeter (P) of a rectangular pen, we need to add up all the sides. In this case, the pen has two lengths (L) and two widths (W), so the equation for the perimeter is:

P = 2L + 2W

b) The area (A) of a rectangle is calculated by multiplying its length (L) by its width (W). In this case, the area is given as 180. So, the equation relating the width and length is:

A = L * W

Since the area is given as 180, we have:

180 = L * W

c) To find the dimensions that require the minimum amount of fencing, we need to minimize the perimeter. To do this, we can use the equation from part (a) and substitute the area equation from part (b).

Given that the area is 180, we have:

180 = L * W

Now, we can solve the area equation for one variable (let's say L), and substitute it into the perimeter equation:

L = 180 / W

P = 2L + 2W

Substituting L with (180 / W):

P = 2(180 / W) + 2W

To find the minimum amount of fencing, we can find the minimum point of the perimeter equation by taking the derivative with respect to W and setting it equal to zero.

Let F(W) = 2(180 / W) + 2W

To find the derivative F'(W), we can use the quotient rule:

F'(W) = (-2 * 180) / W^2 + 2

Setting F'(W) equal to zero:

(-2 * 180) / W^2 + 2 = 0

Simplifying:

-360 / W^2 + 2 = 0

-360 + 2W^2 = 0

2W^2 = 360

W^2 = 180

W = √180 ≈ 13.42

Since the width of a pen cannot be negative, we can ignore the negative value.

Therefore, the width that requires the minimum amount of fencing is approximately 13.42.

To find the corresponding length (L), we can substitute the value of W into the area equation:

180 = L * (13.42)

L = 180 / 13.42 ≈ 13.42

Therefore, the length that requires the minimum amount of fencing is also approximately 13.42.

In summary:
Width = approximately 13.42
Length = approximately 13.42

This means that a square pen with dimensions approximately 13.42 by 13.42 will require the minimum amount of fencing.