I want to create a metal frame of uniform width for a mirror. The mirror is 20 in by 24 in. I bought 416 square in of metal.

Question

Write an expression to find the width of the border.

Solve the equation

How wide can I make the metal frame

Answer 1

This answer can go one of two ways depending on strict interpretation of the condition of uniformity in width of the border of the mirror. The more impressive way for your math teacher will be the also acceptable use of circularly rounded corners on the mirror  border at the corners of the rectangular mirror and is in an ideally mathematical way better too.

 

From an initial expression of 20(2)x + 24(2)x + πx^2 = 416

 

, you solve through the quadratic formula isolating x to be, after simplification, x = {-44 + 4[(121 + 26π)^1/2]}/π inches for the uniform width in inches.

Answer 2

The length of the outside is 24+2x

The width of the outside is 20+2x

 

(24+2x)(20+2x) - 20(24) = 416

480 + 40x + 48x + 4x² - 480 = 416

4x² + 88x - 416 = 0

x² + 22x - 104 = 0

(x+26)(x-4) = 0

x = -26 or x=4

 

The frame will be 4" in width