To find the height of each screen using the Pythagorean theorem, we can rearrange the theorem for our purposes. The theorem states:
\[ c^2 = a^2 + b^2 \]
Where:
- \( c \) is the length of the diagonal (hypotenuse),
- \( a \) is the width (one leg),
- \( b \) is the height (the other leg).
Rearranging gives us:
\[ b^2 = c^2 - a^2 \]
Thus, the height (\( b \)) can be calculated as follows:
\[ b = \sqrt{c^2 - a^2} \]
Now we can calculate the height for each of the specified screens:
Screen A
- Diagonal (c) = 15 inches
- Width (a) = 13 inches
\[ b = \sqrt{15^2 - 13^2} = \sqrt{225 - 169} = \sqrt{56} \approx 7.48 \text{ inches} \]
Screen B
- Diagonal (c) = 22 inches
- Width (a) = 19 inches
\[ b = \sqrt{22^2 - 19^2} = \sqrt{484 - 361} = \sqrt{123} \approx 11.09 \text{ inches} \]
Screen C
- Diagonal (c) = 32 inches
- Width (a) = 28 inches
\[ b = \sqrt{32^2 - 28^2} = \sqrt{1024 - 784} = \sqrt{240} \approx 15.49 \text{ inches} \]
Screen D
- Diagonal (c) = 40 inches
- Width (a) = 35 inches
\[ b = \sqrt{40^2 - 35^2} = \sqrt{1600 - 1225} = \sqrt{375} \approx 19.36 \text{ inches} \]
Summary of Heights
- Screen A: Height ≈ 7.48 inches
- Screen B: Height ≈ 11.09 inches
- Screen C: Height ≈ 15.49 inches
- Screen D: Height ≈ 19.36 inches
Given that Yoselin’s shelf is 17 inches tall, Screens A, B, C, and D would all fit in terms of height. However, the largest allowable screen by height is Screen D with a height of approximately 19.36 inches, which exceeds the shelf height. Therefore, the largest screen that can fit would actually be Screen C, as its height of approximately 15.49 inches is below the shelf height of 17 inches.
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