would cause the points to move away from and then back to their original locations. Is he correct? Explain. Iultiple Choice A pilot in a plane sees a lake at an angle of depression of \( 31^{\circ} \). The lane is cruising at 20,000 feet. How far away is the lake from the plane? \( 17,143 \mathrm{ft} \) \( 23,333 \mathrm{ft} \) B \( 10,301 \mathrm{ft} \)

To solve the problem, we need to use the concept of right triangles and trigonometric functions. The angle of depression from the plane to the lake is given as \( 31^{\circ} \), and the height of the plane is \( 20,000 \) feet. ### Step-by-step Solution: 1. **Understanding the Situation**: - The plane is at a height of \( h = 20,000 \) feet. - The angle of depression \( \theta = 31^{\circ} \) is the angle between the horizontal line from the plane and the line of sight to the lake. 2. **Setting Up the Right Triangle**: - The height of the plane forms one side of a right triangle (the opposite side). - The horizontal distance from the plane to the lake forms the adjacent side. - We can use the tangent function, which relates the angle of a right triangle to the opposite and adjacent sides. 3. **Using the Tangent Function**: \[ \tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} = \frac{h}{d} \] where \( d \) is the distance from the plane to the lake. 4. **Rearranging the Formula**: \[ d = \frac{h}{\tan(\theta)} \] 5. **Substituting the Known Values**: \[ d = \frac{20,000}{\tan(31^{\circ})} \] Now, let's calculate \( d \) using the tangent of \( 31^{\circ} \). ### Calculation: I will calculate \( d \) using the formula derived above. Calculate the value by following steps: - step0: Calculate: \(\frac{20000}{\tan\left(31\right)}\) - step1: Calculate the trigonometric value: \(-45280.055876\) It seems there was an error in the calculation. Let's clarify the calculation for \( d \) again. ### Correct Calculation Steps: 1. **Calculate \( \tan(31^{\circ}) \)**: - First, we need to find the value of \( \tan(31^{\circ}) \). 2. **Substituting Back into the Formula**: - Once we have \( \tan(31^{\circ}) \), we can substitute it back into the formula to find \( d \). Let's perform the calculation for \( \tan(31^{\circ}) \) and then find \( d \). Calculate the value by following steps: - step0: Calculate: \(\frac{20000}{\tan\left(31\right)}\) - step1: Calculate the trigonometric value: \(-45280.055876\) Calculate or simplify the expression \( \tan(31) \). Calculate the value by following steps: - step0: Calculate: \(\tan\left(31\right)\) The calculation for \( \tan(31^{\circ}) \) gives us approximately \( 0.6009 \). Now, we can use this value to find the distance \( d \): ### Step 1: Calculate \( d \) Using the formula: \[ d = \frac{20,000}{\tan(31^{\circ})} \] Substituting the value of \( \tan(31^{\circ}) \): \[ d = \frac{20,000}{0.6009} \] ### Step 2: Perform the Calculation Now, let's calculate \( d \). Calculate the value by following steps: - step0: Calculate: \(\frac{20000}{0.6009}\) - step1: Convert the expressions: \(\frac{20000}{\frac{6009}{10000}}\) - step2: Multiply by the reciprocal: \(20000\times \frac{10000}{6009}\) - step3: Multiply: \(\frac{20000\times 10000}{6009}\) - step4: Multiply: \(\frac{200000000}{6009}\) The calculation gives us: \[ d \approx 33,283.41 \text{ feet} \] ### Conclusion: The distance from the plane to the lake is approximately \( 33,283.41 \) feet. Now, looking at the multiple-choice options provided: - \( 17,143 \text{ ft} \) - \( 23,333 \text{ ft} \) - \( 10,301 \text{ ft} \) None of these options match the calculated distance. Therefore, it seems there may be an error in the options provided or in the interpretation of the problem. If you have any further questions or need additional assistance, feel free to ask!