Question
A boat is heading towards a lighthouse, where Madelyn is watching from a vertical distance of 138138 feet above the water. Madelyn measures an angle of depression to the boat at point AA to be 20, degrees20
∘
. At some later time, Madelyn takes another measurement and finds the angle of depression to the boat (now at point BB) to be 73, degrees73
∘
. Find the distance from point AA to point BB. Round your answer to the nearest tenth of a foot if necessary.
∘
. At some later time, Madelyn takes another measurement and finds the angle of depression to the boat (now at point BB) to be 73, degrees73
∘
. Find the distance from point AA to point BB. Round your answer to the nearest tenth of a foot if necessary.
Answers
GPT-4o mini
To find the distance between points A and B (the locations of the boat at the two angles of depression from Madelyn), we will use trigonometric relationships involving the angles of depression.
1. **Setup the problem:**
- Let \( h = 138138 \) feet (the height of the lighthouse).
- The angle of depression from point A to the boat is \( \theta_1 = 20^\circ \).
- The angle of depression from point B to the boat is \( \theta_2 = 73^\circ \).
2. **Finding distances from the lighthouse to the boat at points A and B:**
- The distance from the base of the lighthouse (point C) to the boat at point A can be represented as \( d_A \).
- The distance from the base of the lighthouse (point C) to the boat at point B can be represented as \( d_B \).
Using the tangent of the angles for the right triangles formed:
\[
\tan(\theta_1) = \frac{h}{d_A} \implies d_A = \frac{h}{\tan(\theta_1)} = \frac{138138}{\tan(20^\circ)}
\]
\[
\tan(\theta_2) = \frac{h}{d_B} \implies d_B = \frac{h}{\tan(\theta_2)} = \frac{138138}{\tan(73^\circ)}
\]
3. **Calculate \( d_A \) and \( d_B \):**
- Calculating \( \tan(20^\circ) \) and \( \tan(73^\circ) \):
- \( \tan(20^\circ) \approx 0.3640 \)
- \( \tan(73^\circ) \approx 3.601 \)
Now substitute these values into the equations for \( d_A \) and \( d_B \):
\[
d_A = \frac{138138}{0.3640} \approx 379779.45 \text{ feet}
\]
\[
d_B = \frac{138138}{3.601} \approx 38320.06 \text{ feet}
\]
4. **Finding the distance from point A to point B:**
The distance between points A and B on the straight line can be calculated as:
\[
d = d_A - d_B = 379779.45 - 38320.06 \approx 341459.39 \text{ feet}
\]
5. **Round to the nearest tenth of a foot:**
\[
d \approx 341459.4 \text{ feet}
\]
Thus, the distance from point A to point B is approximately **341459.4 feet**.
1. **Setup the problem:**
- Let \( h = 138138 \) feet (the height of the lighthouse).
- The angle of depression from point A to the boat is \( \theta_1 = 20^\circ \).
- The angle of depression from point B to the boat is \( \theta_2 = 73^\circ \).
2. **Finding distances from the lighthouse to the boat at points A and B:**
- The distance from the base of the lighthouse (point C) to the boat at point A can be represented as \( d_A \).
- The distance from the base of the lighthouse (point C) to the boat at point B can be represented as \( d_B \).
Using the tangent of the angles for the right triangles formed:
\[
\tan(\theta_1) = \frac{h}{d_A} \implies d_A = \frac{h}{\tan(\theta_1)} = \frac{138138}{\tan(20^\circ)}
\]
\[
\tan(\theta_2) = \frac{h}{d_B} \implies d_B = \frac{h}{\tan(\theta_2)} = \frac{138138}{\tan(73^\circ)}
\]
3. **Calculate \( d_A \) and \( d_B \):**
- Calculating \( \tan(20^\circ) \) and \( \tan(73^\circ) \):
- \( \tan(20^\circ) \approx 0.3640 \)
- \( \tan(73^\circ) \approx 3.601 \)
Now substitute these values into the equations for \( d_A \) and \( d_B \):
\[
d_A = \frac{138138}{0.3640} \approx 379779.45 \text{ feet}
\]
\[
d_B = \frac{138138}{3.601} \approx 38320.06 \text{ feet}
\]
4. **Finding the distance from point A to point B:**
The distance between points A and B on the straight line can be calculated as:
\[
d = d_A - d_B = 379779.45 - 38320.06 \approx 341459.39 \text{ feet}
\]
5. **Round to the nearest tenth of a foot:**
\[
d \approx 341459.4 \text{ feet}
\]
Thus, the distance from point A to point B is approximately **341459.4 feet**.