Question

The length of a rectangular photo is 1 centimeter less than three times its width. If the area of the photo is 102 square centimeters, find the length and width of the
photo.

The length is

and the width is

(Type whole numbers.)

Answer 1

Answers:

• Length = 17
• Width = 6

======================================================

How to get those answers:

Let w be the unknown width in centimeters. This variable is some positive real number. This means w > 0 which will be useful later.

The length is "1 cm less than 3 times its width" and it tells us the length is defined by the expression 3w-1. Whatever w is, we triple it to get 3w and then subtract 1 to get the final length.

Multiply the length and width to get the area 102

length*width = area

(3w-1)*w = 102

3w^2-w = 102

3w^2-w-102 = 0

We could guess and check our way to factoring this, but that's not very efficient. The quadratic formula is the better option. It may seem a bit messy, but it's a more direct path that doesn't involve guessing.

$w = \frac{-b\pm\sqrt{b^2-4ac}}{2a}\\\\w = \frac{-(-1)\pm\sqrt{(-1)^2-4(3)(-102)}}{2(3)}\\\\w = \frac{1\pm\sqrt{1225}}{6}\\\\w = \frac{1\pm35}{6}\\\\w = \frac{1+35}{6}\ \text{ or } \ w = \frac{1-35}{6}\\\\w = \frac{36}{6}\ \text{ or } \ w = \frac{-34}{6}\\\\w = 6\ \text{ or } \ w \approx -5.667$

We ignore the second solution (w = -5.667 approximately) because we stated earlier that w > 0. In other words, a negative length does not make sense, so that's why we ignore it.

-----------------

If w = 6 cm is the width, then 3w-1 = 3*6-1 = 18-1 = 17 cm is the length.

Note that length*width = 17*6 = 102 which is the proper area we want. This confirms the answers.