# Rectangles

The area of a rectangle equals length multiplied by width.

Area = 450

Rectangle questions usually involve algebra and a quadratic equation.

For example, if the length of a rectangle is twice its width, and the rectangle's area is 98, what is the width of the rectangle?

Let w equal width and l equal length.

w(l) = 98 Set up the formula.

w(2w) = 98 Substitute 2l for

${\displaystyle 2w^{2}=98}$ Expand the parentheses.

${\displaystyle w^{2}=49}$ Divide both sides by 2.

w = 7 Take the square root of both sides.

The width of the rectangle is 7 (and the length is 14).

### Practice

1. A rectangle has an area of 132 and its length is 1 greater than its width. What are the dimensions of the rectangle?

2. A rectangle's area would increase by 90 if its length were extended by 18. What is the rectangle's width?

1. 11 and 12

The formula for a rectangle's area is length multiplied by width. Thus, this problem can be solved with algebra and factoring. Let l equal the length and w equal the width.

${\displaystyle l(w)=132}$ Take the initial equation.

${\displaystyle (w+1)(w)=132}$ Substitute the length in terms of the width.

${\displaystyle w^{2}+w=132}$ Expand the parentheses.

${\displaystyle w^{2}+w-132=0}$ Subtract 132 from both sides.

${\displaystyle (w+?)(w+?)=0}$ Break the expression into factors. What two numbers multiply to -132 and add to 1?

${\displaystyle (w+12)(w-11)=0}$ Factor the equation. w equals 11 or -12. Since the width is not negative, it equals 11.

Using the width, the length can be easily determined by adding 1.

2. 5

Since area of a rectangle is length multiplied by width, the width equals the amount of extension increase - 90 - divided by the increase in length, which is 18. The width is thus 5.