Question

the uss nimitz is a 100,000 ton nuclear - powered supercarrier with a runway for fixed - wing fighter jets. suppose a fighter is flying at an altitude of 1020 feet above the flight deck and coming in for a landing with the carrier straight ahead. the angle of depression to the beginning of the runway (touchdown) is 23.8°, and the angle of depression to the end of the runway (splash) is 18.9°. what is the approximate length of the runway?
distance ≈ feet
round your answer to the nearest integer.
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Explanation:

Step1: Define the right triangles

Let the height of the fighter above the flight deck be \( h = 1020 \) feet. Let \( x_1 \) be the horizontal distance from the fighter to the beginning of the runway, and \( x_2 \) be the horizontal distance from the fighter to the end of the runway. The angle of depression to the beginning of the runway is \( \theta_1 = 23.8^\circ \), and to the end is \( \theta_2 = 18.9^\circ \). Since the angle of depression is equal to the angle of elevation from the runway to the fighter (alternate interior angles), we can use the tangent function: \( \tan(\theta) = \frac{h}{x} \), so \( x = \frac{h}{\tan(\theta)} \).

Step2: Calculate \( x_1 \)

For the beginning of the runway (\( \theta_1 = 23.8^\circ \)):
\( x_1 = \frac{1020}{\tan(23.8^\circ)} \)
Using a calculator, \( \tan(23.8^\circ) \approx 0.4407 \), so \( x_1 \approx \frac{1020}{0.4407} \approx 2314.5 \)

Step3: Calculate \( x_2 \)

For the end of the runway (\( \theta_2 = 18.9^\circ \)):
\( x_2 = \frac{1020}{\tan(18.9^\circ)} \)
Using a calculator, \( \tan(18.9^\circ) \approx 0.3417 \), so \( x_2 \approx \frac{1020}{0.3417} \approx 3008.5 \)

Step4: Find the runway length

The length of the runway \( L = x_2 - x_1 \)
\( L \approx 3008.5 - 2314.5 = 694 \)

Answer:

694