Question

A gardener wants to create a rectangular vegetable garden in a backyard. She wants it to have a total area of 120 square feet, and it should be 12 feet longer than it is wide. What dimensions should she use for the vegetable garden? Round to the nearest hundredth of a foot.

Answer 1

Comment
This is an area problem. The key words are 120 square feet and 12 feet longer.
And of course width is a key word when you are reading this.

Formula
Area = L * W

Givens
W = W
L = W + 12

Substitute and Solve
Area = L* W
120 = W*(W + 12)
W^2 + 12W = 120 square feet
w^2 + 12w - 120 = 0

This does not factor easily. I would have thought that a graph might help but not if the dimension has to be to the nearest 1/100 of a foot. The only thing we can do is use the quadratic formula.

a = 1
b = 12
c = - 120

w = [ -b +/- sqrt(b^2 - 4ac) ]/(2a)
w = [-12 +/- sqrt(12^2 - 4*(1)(-120)] / 2*1
w = [-12 +/- sqrt(144 - (-480)]/2
w = [-12 +/- sqrt(624)] / 2
w = [- 12 +/- 24.979992] / 2 The minus root has no meaning whatever.
w = (12.979992) / 2
w = 6.489995 I'll round all this when I get done

L = w + 12
L = 6.489995 + 12
L = 18.489995

check
Area = L * W
Area = 6.489995*18.489995
Area = 119.999935 The difference is a rounding error

Answer
L = 18.489995 = 18.49 feet
W = 6.489995 = 6.49 feet

Note: in the check if you round first to the answer, LW = 120.0001 when you find the area for the check. Kind of strange how that nearest 1/100th makes a difference.