Question

The length of a rectangle is 5 in. more than 3 times

the width. The area of the rectangle is 68 in². What
are the dimensions of the rectangle?
Reminder: The area of a rectangle is given by the
formula A=Iw, where is the measure of the
length and w is the measure of the width.

Answer 1

length = 17 in , width = 4 in

Explanation:

let the width of the rectangle be w

then length is 5 more than 3 times the width, that is 3w + 5

the area (A) of a rectangle is calculated as

A = width × length

given A = 68 in²

substitute values into the formula for A

68 = w(3w + 5) ← distribute parenthesis

68 = 3w² + 5w ( subtract 68 from both sides )

0 = 3w² + 5w - 68 ← quadratic equation in standard form

To factorise the quadratic

Consider the factors of the product of the coefficient of the w² term and the constant term which sum to give the coefficient of the w- term.

product = 3 × - 68 = - 204 and sum = + 5

the factors are - 12 and + 17

use these factor to split the w- term

3w² - 12w + 17w - 68 = 0 ( factor the first/second and third/fourth terms )

3w(w - 4) + 17(w - 4) = 0 ← factor out (w - 4) from each term

(w - 4)(3w + 17) = 0 ← in factored form

equate each factor to zero and solve for w

w - 4 = 0 ⇒ w = 4

3w + 17 = 0 ⇒ 3w = - 17 ⇒ w = -
`(17)/(3)`

however, w > 0 , then w = 4

and length = 3w + 5 = 3(4) + 5 = 12 + 5 = 17

dimensions of rectangle are length = 17 in and width = 4 in