Find the maximum area of the enclosed rectangle using a function.

Question

A family wants to build a fence around a rectangular area of their backyard using their house as one side of the rectangle. The family has $2400 to spend on the fence and is considering using either a fence that costs $50 a yard or a fence that cost $60 a yard.
Suppose the family builds the fence using the $50 yard fence. Let x be the width of the enclosed rectangle. Using a function for the area of the enclosed rectangle, find the maximum area.

Answer 1

 

$2400/$50 = 48

So they have 48 yards for the perimeter of this rectangle.

 

Perimeter = 2 * length + 2* width --> P = 2*L + 2* W

 

BUT.....

 

One side of length is the house, so there is only ONE LENGTH, not two.

So the perimeter formula is, IN THIS PROBLEM ONLY,

 

P = L + 2* w --> P =  L + 2x <--- the independent variable, per the directions in the problem, is to let the width be x.

                    ---> 48 = L + 2x <--- the perimeter is 48 as shown above.

 

Solving for L ---> L = 48 - 2x

 

The area is length x width ---> Area = Length x width ---> A = L x w

                                                                              ----> A = (48 - 2x)*x

 

So the Area function in terms of width x is A(x) = (48-2x)*x 

 

This is parabola with roots 0 = (48-2x)*x

                                       x = 0 or 48 - 2x = 0

                                        x=0 or 48=2x

                                       x=0 or x = 24

 

 The max. occurs at the average of the two roots which is (0 + 24)/2 = 24/2 = 12

 

A(12) = (48 - 2*12)*12 = (48 - 24)*12 = 24*12 = 288

 

The maximum area is 288 square yards. The width is 12 yd. The length is 24 yards. 

The perimeter is 12 + 12 + 24 yd = 48 yd, which at $50 per yard is $2400.

 

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At $60 per yard, 2400 / 60 = 40.

This time they only have 40 yard perimeter with which to work.

 

40 = L + 2x ---> L = 40 -2x.

 

Area = L * W = (40-2x)*x

 

The roots are x=0 and x=20. THe max occurs at (0 +20)/2 = 10.

The max area is 200. The width is x=10. The length is 20.

The perimeter is 10 + 10 + 20 = 40. At $60 per yard, that's $2400.

 

You didn't ask for the $60 part in the problem, but there it is.

The answer to the $50 part in the original problem is highlighted in bold.

I went through the $60 more quickly, since you have seen it once.

Please follow the steps through until you understand.

Contact us if you do not.

Good luck, and thanks for the cool problem. ;-)