A boat is heading towards a lighthouse, where Madeline is watching from a vertical distance of 140 feet above the water. Madeline measures an angle of depression to the boat at point \( A \) to be \( 5^{\circ} \). At some later time, Madeline takes another measurement and finds the angle of depression to the boat (now at point \( B \) ) to be \( 50^{\circ} \). Find the distance from point \( A \) to point \( B \). Round your answer to the nearest tenth of a foot if necessary.
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To find the distance from point \( A \) to point \( B \), we start by using the information given about the angles of depression and Madeline's height above the water. 1. **Finding Distance to Point A**: Madeline is 140 feet above the water, and the angle of depression to the boat at point \( A \) is \( 5^{\circ} \). Using the tangent function: \[ \tan(5^{\circ}) = \frac{140}{d_A} \] where \( d_A \) is the horizontal distance from the base of the lighthouse to point \( A \). Rearranging gives: \[ d_A = \frac{140}{\tan(5^{\circ})} \approx \frac{140}{0.0875} \approx 1600.0 \, \text{feet} \] 2. **Finding Distance to Point B**: Now, for point \( B \), with an angle of depression of \( 50^{\circ} \): \[ \tan(50^{\circ}) = \frac{140}{d_B} \] where \( d_B \) is the horizontal distance to point \( B \). Rearranging gives: \[ d_B = \frac{140}{\tan(50^{\circ})} \approx \frac{140}{1.1918} \approx 117.7 \, \text{feet} \] 3. **Finding the Distance from A to B**: Since both distances are from the base of the lighthouse, the distance between points \( A \) and \( B \) is the difference between \( d_A \) and \( d_B \): \[ \text{Distance from A to B} = d_A - d_B \approx 1600.0 - 117.7 \approx 1482.3 \, \text{feet} \] Hence, the distance from point \( A \) to point \( B \) is approximately **1482.3 feet**.
