Afarmer wants to build a rectangular pen with 80 feet of fencing. the pen will be built against the wall of the barn so one side of the rectangle won’t need a fence. what dimensions will maximize the area of the pen?

The length of the rectangular pen is

The width of the rectangular pen is

Step-by-step explanation:

Let

x-----> the length of the rectangular pen

y----> the width of the rectangular pen

we know that

The perimeter of the rectangular pen in this problem is equal to

---> remember that one side of the rectangle won’t need a fence

so

-----> equation A

The area of the rectangular pen is equal to

-----> equation B

Substitute equation A in equation B

The quadratic equation is a vertical parabola open downward

The vertex is a maximum

The x-coordinate of the vertex is the length of the rectangular pen that maximize the area of the pen

The y-coordinate of the vertex is the maximum area of the pen

using a graphing tool

The vertex is the point (40,800)

see the attached figure

so

Find the value of y

---->

therefore

The length of the rectangular pen is

The width of the rectangular pen is

80π/3 or about 83.775

step-by-step explanation:

since a cone is 1/3 of a cylinder, we need to find the base's area. since area = πr^2, we get 16π. then, multiplying this by 5, we get 80π. dividing by 3, since the cone's volume is 1/3 of that of a cylinder's, we get 80π/3.

hope this !