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Abby is buying a rectangular widescreen TV that she will hang on the wall between two windows such that the longer side of the TV is horizontal. The windows are 36 inches apart horizontally, and a widescreen TV is approximately twice as wide as it is tall. Which of the following could be the diagonal length of a widescreen TV that fits between the windows? Choose 1 answer: (A) 32 inches (B) 42 inches (C) 55 inches (D) 60 inches

Abby is buying a rectangular widescreen TV that she will hang on the wall between two windows such that the longer side of the TV is horizontal. The windows are 3636 inches apart horizontally, and a widescreen TV is approximately twice as wide as it is tall. Which of the following could be the diagonal length of a widescreen TV that fits between the windows?\newlineChoose 11 answer:\newline(A) 3232 inches\newline(B) 4242 inches\newline(C) 5555 inches\newline(D) 6060 inches

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Q. Abby is buying a rectangular widescreen TV that she will hang on the wall between two windows such that the longer side of the TV is horizontal. The windows are 3636 inches apart horizontally, and a widescreen TV is approximately twice as wide as it is tall. Which of the following could be the diagonal length of a widescreen TV that fits between the windows?\newlineChoose 11 answer:\newline(A) 3232 inches\newline(B) 4242 inches\newline(C) 5555 inches\newline(D) 6060 inches
  1. Understand Problem: Understand the problem and the given information.\newlineThe space between the windows is 3636 inches, and the TV's aspect ratio is approximately 2:12:1, meaning the width is twice the height. We need to find a TV size that fits within this space, considering the diagonal length.
  2. Denote TV Dimensions: Let's denote the height of the TV as hh and the width as ww. According to the aspect ratio, w=2hw = 2h. The diagonal dd can be found using the Pythagorean theorem, where d2=h2+w2d^2 = h^2 + w^2.
  3. Substitute in Pythagorean Theorem: Substitute the width in terms of height into the Pythagorean theorem.\newlined2=h2+(2h)2d^2 = h^2 + (2h)^2\newlined2=h2+4h2d^2 = h^2 + 4h^2\newlined2=5h2d^2 = 5h^2
  4. Set Width Constraint: Since the width of the TV must be less than or equal to 3636 inches to fit between the windows, we set w≤36w \leq 36 inches. Therefore, 2h≤362h \leq 36 inches, which implies h≤18h \leq 18 inches.
  5. Calculate Maximum Diagonal Length: Calculate the maximum possible diagonal length using the maximum height of 1818 inches.d2=5h2d^2 = 5h^2d2=5(18)2d^2 = 5(18)^2d2=5(324)d^2 = 5(324)d2=1620d^2 = 1620d=1620d = \sqrt{1620}d≈40.25 inchesd \approx 40.25 \text{ inches}
  6. Compare with Options: Compare the calculated diagonal length with the given options.\newlineThe calculated diagonal length is approximately 40.2540.25 inches, which is not an option. However, it is closest to option (B) 4242 inches, which is slightly larger than the calculated length. Since the TV must fit between the windows, the diagonal length must be less than or equal to the space available. Therefore, option (A) 3232 inches is the only possible size that is less than the calculated diagonal and also less than the space available between the windows.

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