Question

Charlene is a video game designer and wants to make sure that her games can be played on all types of screens. In order for the games to work on both devices with a standard aspect ratio and those with widescreen aspect ratios, the playable area of the games should have an aspect ratio of 3:2. This means that the width is equal to times the height.
On one type of tablet, the playable area of the game on the screen is 54 square inches. Write and solve a system of equations to determine the dimensions of the playable area.

A. The playable area has a width of 6 inches and a height of 4 inches.

B. The playable area has a width of 6 inches and a height of 9 inches.

C. The playable area has a width of 9 inches and a height of 6 inches.

D. The playable area has a width of 13.5 inches and a height of 9 inches.

Answer 1

Answer:

C. The playable area has a width of 9 inches and a height of 6 inches.

Explanation:

There are a number of ways you can get there.

1. Check the answers to see which have the right area and aspect ratio.

• A: area = 24 in² — does not match 54 in²
• B: area = 54 in², aspect ratio 6:9 = 2:3 — does not match 3:2 aspect ratio
• C: area = 54 in², aspect ratio 9:6 = 3:2 — matches problem statement
• D: area = 121.5 in² — does not match 54 in²

2. If the screen were 3:2 (inches), its area would be 6 in². The area of 54 in² is 9 times that value, so the actual screen dimensions are √9 = 3 times 3:2. That is, they are width:height = 9:6 inches — matches selection C.

3. You can write equations for width and height and solve.

• w/h = 3/2
• wh = 54

Substituting w=3/2·h into the second equation gives

... (3/2)h·h = 54

... h² = 36 . . . . . multiply by 2/3

... h = √36 = 6 . . . . square root, result in inches . . . . matches selection C