The area of a rectangular piece of cardboard is 90 square centimeters, and the perimeter is 46 centimeters. How do you find the dimensions of the rectangle?

1 Answer
Douglas K.
Nov 14, 2016

Please see the explanation.

Explanation:

Let L = the length

Let W = the width

#LW = 90" cm"^2" [1]"#

#2L + 2W = 46" cm [2]"#

Divide equation [2] by 2:

#L + W = 23" cm"#

Subtract L from both sides:

#W = 23" cm" - L#

Substitute #23" cm" - L# for W in equation [1]:

#L(23" cm" - L) = 90" cm"^2#

Use the distributive property

#23" cm"(L) - L^2 = 90" cm"^2#

Subtract #90" cm"^2# from both sides:

#23" cm"(L) - L^2 - 90" cm"^2 = 0#

Multiply both sides by -1:

#L^2 - 23" cm"(L) + 90" cm"^2 = 0#

Having solved this type of problem with the quadratic formula, many times, I know that the greater of the two solutions gives the length and the lesser, the width:

#L = (23" cm" + sqrt((23" cm")^2 - 4(1)(90" cm"^2)))/2#

#L = 18" cm"#

#W = (23" cm" - sqrt((23" cm")^2 - 4(1)(90" cm"^2)))/2#

#W = 5" cm"#

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