Question

Computer screens and television screens are usually measured by the diagonal of the screen. For example, a screen of size 60 inches would have a diagonal of 60 inches, a width of 52 inches, and a height of 30 inches. Yoselin would like to install a new screen on a shelf that is 17 inches tall and 50 inches wide. She would like the screen to be the largest size that will fit on the shelf. She knows the diagonal and width of the 4 screens listed below. Screen Diagonal Width Height A 15” 13” B 22” 19” C 32” 28” D 40” 35” 1. Use the Pythagorean Theorem to calculate the height of each screen, and complete the table.

Answer 1

To find the height of each screen using the Pythagorean Theorem, we can use the relationship between the width, height, and diagonal of the screen, given by the formula:

\[
d^2 = w^2 + h^2
\]

where:
- \( d \) is the diagonal,
- \( w \) is the width, and
- \( h \) is the height.

We can rearrange this formula to solve for height \( h \):

\[
h = \sqrt{d^2 - w^2}
\]

Now, we will calculate the height of each screen based on the data provided:

1. **Screen A:**
- Diagonal \( d = 15” \)
- Width \( w = 13” \)
\[
h = \sqrt{15^2 - 13^2} = \sqrt{225 - 169} = \sqrt{56} \approx 7.48”
\]

2. **Screen B:**
- Diagonal \( d = 22” \)
- Width \( w = 19” \)
\[
h = \sqrt{22^2 - 19^2} = \sqrt{484 - 361} = \sqrt{123} \approx 11.09”
\]

3. **Screen C:**
- Diagonal \( d = 32” \)
- Width \( w = 28” \)
\[
h = \sqrt{32^2 - 28^2} = \sqrt{1024 - 784} = \sqrt{240} \approx 15.49”
\]

4. **Screen D:**
- Diagonal \( d = 40” \)
- Width \( w = 35” \)
\[
h = \sqrt{40^2 - 35^2} = \sqrt{1600 - 1225} = \sqrt{375} \approx 19.36”
\]

Now, we can summarize the results in a table:

| Screen | Diagonal (inches) | Width (inches) | Height (inches) |
|--------|--------------------|----------------|------------------|
| A | 15 | 13 | ≈ 7.48 |
| B | 22 | 19 | ≈ 11.09 |
| C | 32 | 28 | ≈ 15.49 |
| D | 40 | 35 | ≈ 19.36 |

Based on the values above, Yoselin can now determine which screen fits within the dimensions of her shelf (17 inches tall and 50 inches wide). All calculated heights fit under 17 inches. The largest screen will be screen D, which has the maximum height of approximately 19.36 inches, but, since it exceeds the shelf height, screen C with a height of approximately 15.49 inches would likely be the largest screen she can choose.