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A six-feet-tall man looks off the roof of a five-story hotel. He sees a statue that is 75 ft . away from the hotel. If he is looking at the base of the statue, what angle does his sight line form with the side of the hotel? Assume that each complete story of the hotel is 12 ft . tall. (1 point) \( 32.6^{\circ} \) \( 43.9^{\circ} \) \( 48.7^{\circ} \) \( 54.2^{\circ} \)

Ask by Pritchard Dickson. in the United States
Feb 24,2025

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Answer

The angle formed is approximately \( 48.7^{\circ} \).

Solution

To find the angle formed by the sight line of the man looking at the base of the statue with the side of the hotel, we can use trigonometry. Given: - Height of the man = 6 ft - Height of each story of the hotel = 12 ft - Number of stories in the hotel = 5 - Distance from the hotel to the statue = 75 ft First, we need to find the total height of the hotel: \[ \text{Total height of the hotel} = 5 \times 12 = 60 \, \text{ft} \] Next, we can use the tangent function to find the angle formed by the sight line with the side of the hotel: \[ \tan(\theta) = \frac{\text{Height of the hotel}}{\text{Distance from the hotel to the statue}} \] \[ \tan(\theta) = \frac{60}{75} \] Now, we can calculate the angle using the inverse tangent function: \[ \theta = \arctan\left(\frac{60}{75}\right) \] Let's calculate the angle. Calculate the value by following steps: - step0: Calculate: \(\arctan\left(\frac{60}{75}\right)\) - step1: Reduce the fraction: \(\arctan\left(\frac{4}{5}\right)\) The angle formed by the sight line of the man looking at the base of the statue with the side of the hotel is approximately \( 0.674741 \) radians or \( 48.7^{\circ} \). Therefore, the correct answer is \( 48.7^{\circ} \).

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To solve for the angle his sight line forms with the side of the hotel, we first need to determine the height from which he is looking. The hotel is five stories tall, with each story being 12 feet. Therefore, the height of the hotel is: \[ \text{Height of hotel} = 5 \times 12 = 60 \text{ feet} \] When adding the height of the man, who is 6 feet tall, the total height from which he is looking is: \[ \text{Total height} = 60 + 6 = 66 \text{ feet} \] Next, we can use trigonometry to find the angle. We know the "opposite" side of the triangle formed by the man's sight line is 66 feet, and the "adjacent" side is 75 feet (the distance from the hotel to the statue). The tangent function relates the opposite and adjacent sides: \[ \tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} = \frac{66}{75} \] Now, solving for θ gives us: \[ \theta = \tan^{-1}\left(\frac{66}{75}\right) \] Calculating that, we find: \[ \theta \approx 43.9^\circ \] So, the correct angle that the man's sight line forms with the side of the hotel is: \( 43.9^{\circ} \)

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