January 2010
Problem of the Month

HD TV Problem

Isaac told Mr. P that his family had a 106 inch HDTV screen in his basement.

The screen aspect ratio, that is, the ratio of the horizontal length of a TV screen to its vertical height, is 16 : 9 for HD TVs.

By contrast, Mr. P has a 19 inch traditional TV in his bedroom.

Traditional televisions have a 4 : 3 screen aspect ratio.

The "inch size" of the TV always refers to the diagonal measurement of the TV, so the 106" and 19" above would be the diagonal measurements.

Determine the viewing area of each TV!
You may round the horizontal and vertical lengths to the nearest inch before finding the areas.

Solution to the Problem:

Isaac's family TV has 4,784 square inches while Mr. P's TV has 165 square inches.   (using the rounded values for the horizontal and vertical lengths)

Let 16x and 9x be the horizontal and vertical lengths, respectively, of Isaac's TV.

Then using the Pythagorean Theorem, (16x)² + (9x)² = 106²
Solving, we obtain 337 x² = 11236,
so x = 5.7741.
Therefore, 16x = 16 (5.7741) = 92.39 or 92 inches.

Likewise, 9x = 9 (5.7741) = 51.97 or 52 inches.
The area is 52 x 92 = 4,784 square inches.

In a similar manner, let 4x and 3x be the horizontal and vertical lengths, respectively, of Mr. P's TV.

Then using the Pythagorean Theorem, (4x)² + (3x)² = 19²

Solving, we obtain 25 x² = 361,
so x = 3.8.
Therefore, 4x = 4 (3.8) = 15.2 or 15 inches.

Likewise, 3x = 3 (3.8) = 11.4 or 11 inches.
The area is 15 x 11 = 165 square inches.

Correctly solved by:

1. Gusti Oggenfuss

Montet (Broye), Suisse (Switzerland)

Send any comments or questions to: David Pleacher