SOLUTION: A dog trainer has 100 ft of fencing that will be used to create a rectangular work area for dogs. If the trainer wants to enclose an area of 504 ft2, what will be the dimensions of

Question 1202100: A dog trainer has 100 ft of fencing that will be used to create a rectangular work area for dogs. If the trainer wants to enclose an area of 504 ft2, what will be the dimensions of the work area?
Found 3 solutions by josgarithmetic, math_tutor2020, ikleyn:
Answer by josgarithmetic(39620)   (Show Source): You can put this solution on YOUR website!
Use all the 100 feet fence material?

w width
L length

-

-
discriminant

OR

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Dimensions be 14 feet by 36 feet.
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Answer by math_tutor2020(3817)   (Show Source): You can put this solution on YOUR website!

Answer: 14 ft by 36 ft

Work Shown:

x = width
y = height

P = perimeter of the rectangle
P = 2*(width + height)
P = 2*(x + y)
Replace P with 100 as this is the perimeter, aka amount of fencing
100 = 2(x + y)
100/2 = x + y
50 = x + y
y = 50-x

A = area of the rectangle
A = width*height
A = x*y
A = x*(50-x)
A = 50x-x^2
A = -x^2+50x
Plug in A = 504 which is the desired area.
504 = -x^2+50x
0 = -x^2+50x-504
-x^2+50x-504 = 0
-(x^2-50x+504) = 0
x^2-50x+504 = 0

You could take a trial-and-error approach to factoring, but I think the quadratic formula is more efficient.
Plug in a = 1, b = -50, c = 504

or

or

or

If x = 36, then y = 50-x = 50-36 = 14
If x = 14, then y = 50-x = 50-14 = 36

The rectangle is 14 ft by 36 ft

Check:
Perimeter = 2*(14+36) = 2*50 = 100
Area = 14*36 = 504
The answers are confirmed.

Answer by ikleyn(52803)   (Show Source): You can put this solution on YOUR website!
.
A dog trainer has 100 ft of fencing that will be used to create a rectangular work area for dogs.
If the trainer wants to enclose an area of 504 ft2, what will be the dimensions of the work area?
~~~~~~~~~~~~~~~~~

L + W should be 100/2 = 50 ft. So, if x is the width (in feet), then the length is (50-x) ft. The area equation is x*(50-x) = 504 50x - x^2 = 504 x^2 - 50x + 504 = 0 Solve using the quadratic formula = = . The roots are = = 36 and = 14. They produce the same rectangle with dimensions 36 ft and 14 ft. ANSWER. The dimensions of the rectangle are 36 ft by 14 ft.

Solved.

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