To find the height of each screen, we can use the Pythagorean theorem, which states that for a right triangle:
\[ c^2 = a^2 + b^2 \]
Where:
- \( c \) is the diagonal (the longest side),
- \( a \) is the width,
- \( b \) is the height.
We will rearrange the formula to find the height \( b \):
\[ b = \sqrt{c^2 - a^2} \]
Now, let's calculate the height for each television screen based on the provided data.
Data Table:
-
Screen A:
- Diagonal \( c = 15 \)
- Width \( a = 13 \)
\[ b = \sqrt{15^2 - 13^2} \] \[ b = \sqrt{225 - 169} \] \[ b = \sqrt{56} \] \[ b \approx 7.5 \]
-
Screen B:
- Diagonal \( c = 22 \)
- Width \( a = 19 \)
\[ b = \sqrt{22^2 - 19^2} \] \[ b = \sqrt{484 - 361} \] \[ b = \sqrt{123} \] \[ b \approx 11.1 \]
-
Screen C:
- Diagonal \( c = 32 \)
- Width \( a = 28 \)
\[ b = \sqrt{32^2 - 28^2} \] \[ b = \sqrt{1024 - 784} \] \[ b = \sqrt{240} \] \[ b \approx 15.5 \]
-
Screen D:
- Diagonal \( c = 40 \)
- Width \( a = 35 \)
\[ b = \sqrt{40^2 - 35^2} \] \[ b = \sqrt{1600 - 1225} \] \[ b = \sqrt{375} \] \[ b \approx 19.4 \]
Summary of Heights:
- Screen A: Height ≈ 7.5 inches
- Screen B: Height ≈ 11.1 inches
- Screen C: Height ≈ 15.5 inches
- Screen D: Height ≈ 19.4 inches
Finding the Biggest TV:
Now we need to determine the largest diagonal that will fit on the 17 x 50 inch shelf. The largest allowable diagonal (hypotenuse) when fitting TVs onto the shelf can be found with the Pythagorean theorem:
Given the shelf dimensions:
- Width \( a = 17 \)
- Height \( b = 50 \)
Find the diagonal \( c \):
\[ c = \sqrt{17^2 + 50^2} \] \[ c = \sqrt{289 + 2500} \] \[ c = \sqrt{2789} \] \[ c \approx 52.8 \]
Conclusion:
All four screens fit on the shelf because their diagonals (15, 22, 32, 40) are less than the computed maximum diagonal of approximately 52.8 inches. Thus, the largest TV that fits would be the one with a diagonal of 40 inches (Screen D).
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