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A boat is heading towards a lighthouse, where Dalvin is watching from a vertical distance of $138$ feet above the water. Dalvin measures an angle of depression to the boat at point $A$ to be $13$ degrees. At some later time, Dalvin takes another measurement and finds the angle of depression to the boat (now at point $B$ ) to be $45$ degrees. Find the distance from point $A$ to point $B$ .

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Evaluate two-variable equations: word problems

Full solution

Q. A boat is heading towards a lighthouse, where Dalvin is watching from a vertical distance of

138

feet above the water. Dalvin measures an angle of depression to the boat at point

A

to be

13

degrees. At some later time, Dalvin takes another measurement and finds the angle of depression to the boat (now at point

B

) to be

45

degrees. Find the distance from point

A

to point

B

.

Identify Given Information: Step $1$ : Identify the information given and the formula needed. $\newline$ Dalvin is $138$ feet above the water, and the angles of depression to the boat at points $A$ and $B$ are $13$ degrees and $45$ degrees, respectively. We use the tangent of the angle of depression to find the horizontal distances from Dalvin to points $A$ and $B$ .
Calculate Distance to Point A: Step $2$ : Calculate the horizontal distance from Dalvin to point A using the tangent of $13$ degrees. $\newline$ Distance $A = \frac{138}{\tan(13^\circ)}$ $\newline$ Distance $A \approx \frac{138}{0.2309} \approx 598$ feet
Calculate Distance to Point B: Step $3$ : Calculate the horizontal distance from Dalvin to point B using the tangent of $45$ degrees. $\newline$ Distance B = $138 / \tan(45°)$ $\newline$ Distance B $\approx 138 / 1 \approx 138$ feet
Find Distance Between Points A and B: Step $4$ : Find the distance between point A and point B. $\newline$ Distance from A to B = Distance A - Distance B $\newline$ Distance from A to B = $598 - 138 = 460$ feet

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