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A boat is heading towards a lighthouse, whose beacon-light is 113113 feet above the water. The boat’s crew measures the angle of elevation to the beacon, 11∘11^\circ. What is the ship’s horizontal distance from the lighthouse (and the shore)?

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Full solution

Q. A boat is heading towards a lighthouse, whose beacon-light is 113113 feet above the water. The boat’s crew measures the angle of elevation to the beacon, 11∘11^\circ. What is the ship’s horizontal distance from the lighthouse (and the shore)?
  1. Understand and Visualize: Understand the problem and visualize the scenario.\newlineWe have a right triangle where the lighthouse is the vertical side (opposite to the angle of elevation), the horizontal distance from the boat to the lighthouse is the adjacent side, and the angle of elevation is given as 1111 degrees.
  2. Identify Trigonometric Function: Identify the trigonometric function to use.\newlineTo find the horizontal distance (adjacent side), we use the cosine function, which relates the adjacent side to the hypotenuse in a right triangle.\newlinecos⁡(θ)=adjacenthypotenuse\cos(\theta) = \frac{\text{adjacent}}{\text{hypotenuse}}
  3. Set up Equation: Set up the equation using the cosine function. cos⁡(11∘)=horizontal distance113 feet\cos(11^\circ) = \frac{\text{horizontal distance}}{113 \text{ feet}}
  4. Solve for Distance: Solve for the horizontal distance.\newlinehorizontal distance = 113 feet×cos⁡(11°)113 \text{ feet} \times \cos(11°)\newlineNow we calculate the cosine of 1111 degrees and multiply it by 113113 feet.
  5. Calculate Cosine and Multiply: Calculate the cosine of 1111 degrees and multiply by 113113 feet.\newlineUsing a calculator, we find that cos⁡(11∘)≈0.9816\cos(11^\circ) \approx 0.9816.\newlineTherefore, horizontal distance ≈113\approx 113 feet ×0.9816\times 0.9816
  6. Perform Multiplication: Perform the multiplication to find the horizontal distance. \newlinehorizontal distance ≈113 feet×0.9816≈110.9408 feet\approx 113 \text{ feet} \times 0.9816 \approx 110.9408 \text{ feet}
  7. Round Answer: Round the answer to a reasonable degree of precision.\newlineSince the angle was given to two significant figures, we can round the horizontal distance to the nearest foot.\newlinehorizontal distance ≈111\approx 111 feet

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