Question

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?

Answer 1

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"`