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Makayla spots an airplane on radar that is currently approaching in a straight line, and that fly directly overhead. The plane maintains a constant altitude of 5750 feet. Makayla initially measures an angle of elevation of 16^(@) to the plane at point A. At some later time, she measur an angle of elevation of 31^(@) to the plane at point B. Find the distance the plane traveled from point A to point B. Round your answer to the nearest foot if necessary.

Makayla spots an airplane on radar that is currently approaching in a straight line, and that fly directly overhead. The plane maintains a constant altitude of 57505750 feet. Makayla initially measures an angle of elevation of 16∘ 16^{\circ} to the plane at point A A . At some later time, she measur an angle of elevation of 31∘ 31^{\circ} to the plane at point B B . Find the distance the plane traveled from point A A to point B B . Round your answer to the nearest foot if necessary.

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Q. Makayla spots an airplane on radar that is currently approaching in a straight line, and that fly directly overhead. The plane maintains a constant altitude of 57505750 feet. Makayla initially measures an angle of elevation of 16∘ 16^{\circ} to the plane at point A A . At some later time, she measur an angle of elevation of 31∘ 31^{\circ} to the plane at point B B . Find the distance the plane traveled from point A A to point B B . Round your answer to the nearest foot if necessary.
  1. Trigonometry Explanation: To solve this problem, we will use trigonometry, specifically the tangent function, which relates the angle of elevation to the opposite side (the altitude of the plane) and the adjacent side (the distance from Makayla to the point directly below the plane). We will calculate the distance from Makayla to the point directly below the plane at both angles of elevation and then find the difference between these two distances to determine the distance the plane traveled from point AA to point BB.
  2. Calculate D11 at 1616 degrees: First, let's calculate the distance from Makayla to the point directly below the plane when the angle of elevation is 1616 degrees. We will call this distance D1D1. We can use the tangent function:\newlinetan⁡(16 degrees)=altitudeD1\tan(16 \text{ degrees}) = \frac{\text{altitude}}{D1}\newlinetan⁡(16 degrees)=5750 feetD1\tan(16 \text{ degrees}) = \frac{5750 \text{ feet}}{D1}\newlineD1=5750 feettan⁡(16 degrees)D1 = \frac{5750 \text{ feet}}{\tan(16 \text{ degrees})}
  3. Calculate D11: Now, we calculate D11 using a calculator:\newlineD1≈5750D1 \approx 5750 feet / tan⁡(16∘)\tan(16^\circ)\newlineD1≈5750D1 \approx 5750 feet / 0.2867450.286745 (approximate value of tan⁡(16∘)\tan(16^\circ))\newlineD1≈20058.7D1 \approx 20058.7 feet
  4. Calculate D2D_2 at 3131 degrees: Next, we calculate the distance from Makayla to the point directly below the plane when the angle of elevation is 3131 degrees. We will call this distance D2D_2. Again, we use the tangent function:\newlinetan⁡(31 degrees)=altitudeD2\tan(31 \text{ degrees}) = \frac{\text{altitude}}{D_2}\newlinetan⁡(31 degrees)=5750 feetD2\tan(31 \text{ degrees}) = \frac{5750 \text{ feet}}{D_2}\newlineD2=5750 feettan⁡(31 degrees)D_2 = \frac{5750 \text{ feet}}{\tan(31 \text{ degrees})}
  5. Calculate D22: Now, we calculate D2D_2 using a calculator:\newlineD2≈5750 feet/tan⁡(31 degrees)D_2 \approx 5750 \text{ feet} / \tan(31 \text{ degrees})\newlineD2≈5750 feet/0.600861D_2 \approx 5750 \text{ feet} / 0.600861 (approximate value of tan⁡(31 degrees)\tan(31 \text{ degrees}))\newlineD2≈9569.4 feetD_2 \approx 9569.4 \text{ feet}
  6. Find Distance Traveled: Finally, we find the distance the plane traveled from point A to point B by subtracting D2D_2 from D1D_1: \newlineDistance traveled = D1−D2D_1 - D_2 \newlineDistance traveled ≈20058.7\approx 20058.7 feet −9569.4- 9569.4 feet \newlineDistance traveled ≈10489.3\approx 10489.3 feet
  7. Round to Nearest Foot: We round the answer to the nearest foot as instructed:\newlineDistance traveled ≈10489\approx 10489 feet

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