# 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?

Nov 14, 2016

#### Explanation:

Let L = the length

Let W = the width

$L W = 90 \text{ cm"^2" }$

$2 L + 2 W = 46 \text{ cm }$

Divide equation  by 2:

$L + W = 23 \text{ cm}$

Subtract L from both sides:

$W = 23 \text{ cm} - L$

Substitute $23 \text{ cm} - L$ for W in equation :

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

Use the distributive property

$23 {\text{ cm"(L) - L^2 = 90" cm}}^{2}$

Subtract $90 {\text{ cm}}^{2}$ from both sides:

$23 {\text{ cm"(L) - L^2 - 90" cm}}^{2} = 0$

Multiply both sides by -1:

${L}^{2} - 23 {\text{ cm"(L) + 90" cm}}^{2} = 0$

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

$L = 18 \text{ cm}$

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

$W = 5 \text{ cm}$