Area of a Rectangle

Question

The length of a rectangle is 7 inches more than 2 times the width. If the area of the rectangle is 22 square inches, find the length and the width.

Answer 1

The area of a rectangle is found by multiplying the length times the width.

The first sentence tells us that

L = 2 W + 7

So we know that the area is 22 and therefore

W (2W A+ 7) = 22

Multiplying this out and rearranging the terms gives

2 W² + 7 W = 22

2 W² + 7 W - 22 = 0

This is factorable by the following technique.

1. multiply the coefficient of the squared term by the constant. 2 X -22 = -44
2. Make a list of all the factors of 44. They are 1 and 44, 2 and 22, and 4 and 11.
3. Now choose the pair so that when you make one of them negative, they add up to the middle coefficient of +7.
4. The correct pair will be -4 and 11.
5. Now rewrite the trinomial putting these two terms in place of the middle term.

2 W² - 4 W + 11 W - 22.

Now factor by grouping

(2 W² - 4 W) + (11 W - 22)

2 W (W - 2) + 11 (W - 2)

(2W + 11) ( W - 2)

Set each factor = 0 and solve.

So W = -11/2 or W = 2

Since a negative width is not acceptable, discard that answer.

So W =2. and L = 2(2) + 7 = 11

The rectangle is 11 by 2 for an area of 22.

Answer 2

Area = L × W = 22 in² and L = 2W +7

Replace L with 2W +7 in the area equation to get

( 2W +7) × W = 22 which means that 2W² +7W = 22

You can find W by solving the quadratic equation 2W² + 7W – 22 = 0

You can solve a quadratic equation by factoring (if possible)

or using the quadratic formula x = (–b ±√(b² – 4ac))/2a

In this case using the quadratic formula is probably quicker

Using the Quadratic Formula

Using the quadratic formula for 2W² + 7W – 22 = 0 you would calculate

W = (–7 ±√(7² – 4(2)(–22)))/2(2) = (–7 ±√(49 +176))/4 = (–7 ±√(225))/4 = (–7 ±15)/4

So you two solutions would be

1. W = (–7 +15)/4 = 8/4 = 2
2. W = (–7 – 15)/4 = –22/4 = –11/2

Since width can't be negative we discard the solution –11/2

and the first part of our answer is W = 2

To find length you just plug W = 2 into the equation L = 2W +7

Hope this helps. Just post a comment if you have any questions.