Question

while sailing a boat offshore, jose sees a lighthouse and calculates that the angle of elevation to the top of the lighthouse is 3°, as shown in the diagram below. when she sails her boat 700 feet closer to the lighthouse, she finds that the angle of elevation is now 5°. how tall is the lighthouse?

Explanation:

Step1: Set up equations using tangent function

Let the height of the lighthouse be \(h\) feet. Let the initial horizontal distance from the boat to the base of the lighthouse be \(x\) feet.
We know that \(\tan\theta=\frac{\text{opposite}}{\text{adjacent}}\).
For the first - angle of elevation (\(\theta = 3^{\circ}\)): \(\tan3^{\circ}=\frac{h}{x}\), so \(x = \frac{h}{\tan3^{\circ}}\).
For the second - angle of elevation (\(\theta = 5^{\circ}\)): \(\tan5^{\circ}=\frac{h}{x - 700}\), so \(x-700=\frac{h}{\tan5^{\circ}}\).

Step2: Substitute \(x\) from the first equation into the second equation

Substitute \(x=\frac{h}{\tan3^{\circ}}\) into \(x - 700=\frac{h}{\tan5^{\circ}}\).
We get \(\frac{h}{\tan3^{\circ}}-700=\frac{h}{\tan5^{\circ}}\).
Since \(\tan3^{\circ}\approx0.0524\) and \(\tan5^{\circ}\approx0.0875\).
\(\frac{h}{0.0524}-700=\frac{h}{0.0875}\).

Step3: Solve for \(h\)

Multiply through by \(0.0524\times0.0875\) to clear the fractions:
\(0.0875h-700\times0.0524\times0.0875 = 0.0524h\).
\(0.0875h-0.0524h=700\times0.0524\times0.0875\).
\((0.0875 - 0.0524)h=700\times0.0524\times0.0875\).
\(0.0351h=700\times0.0524\times0.0875\).
\(h=\frac{700\times0.0524\times0.0875}{0.0351}\).
\(h=\frac{700\times0.00458}{0.0351}\).
\(h=\frac{3.206}{0.0351}\approx91.5\).

Answer:

91.5 ft