I need to figure out the area of this kennel maximized can you look at my description and help me?

Question

a dog kennel owner has 200 feet of fencing available for constructing 3 rectangular regions as shown below. a) write the area of the entire region as a function of one variable

b) what dimensions of the entire region will maximize the area of the region

c) what is the maximum area of the entire region 

 

It gives me a picture of a rectangle broken into 3 smaller rectangles inside of it? 

Answer 1

By the description you've given, you have a rectangle with two parallel sides, which I'll call the lengths L, and four parallel sides (two on the inside), which I'll call widths W.

 

The total perimeter of this shape (including the two interior sides) is

P = 2L + 4W

 

According to the problem, the total perimeter is 200 ft, so

200 = 2L + 4W   --divide through by 2

100 = L + 2W     --solve for L

100 - 2W = L

 

The area of the whole rectangle is

A = L*W    --substitute the previous expression in for L

A = (100 - 2W)*W

A = 100W - 2W²

This is your answer for part a).

 

b) This equation represents a downward facing parabola, so it's vertex will be the maximum value. To find the W-coordinate of the parabola, you can use the shortcut formula

W = -b/2a

W = -100/2(-2)

W = -100/-4

W = 25 ft

 

Substitute this into our expression for L

L = 100 - 2W

L = 100 - 2(25)

L = 50 ft

 

c) The area is then

A = L*W

A = (50 ft)(25 ft)

A = 1250 ft²