Two boats are anchored in still water, along a straight line from a lighthouse of height $41$ meters. From the beacon on top of the lighthouse, the angle to the nearer boat is $43$ degrees and the angle to the farther boat is $67$ . $6$ degrees. Determine the distance that the light travels from the lighthouse to each boat, and the distance between the two boats. $\newline$ Use the diagram to determine the measure of the sought sides. $\newline$ What is the distance between the two boats? $\newline$ meters

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Q. Two boats are anchored in still water, along a straight line from a lighthouse of height

41

meters. From the beacon on top of the lighthouse, the angle to the nearer boat is

43

degrees and the angle to the farther boat is

67

6

degrees. Determine the distance that the light travels from the lighthouse to each boat, and the distance between the two boats.

\newline

Use the diagram to determine the measure of the sought sides.

\newline

What is the distance between the two boats?

\newline

meters

Identify triangles formed: Identify the triangles formed by the lighthouse and the boats. $\newline$ The lighthouse and the two boats form two right-angled triangles with the lighthouse as the common vertex. The height of the lighthouse is one leg of both triangles, and the distances from the lighthouse to each boat are the hypotenuses of the respective triangles.
Calculate distance to nearer boat: Calculate the distance from the lighthouse to the nearer boat using the tangent function. $\newline$ The tangent of the angle to the nearer boat is equal to the opposite side (height of the lighthouse) over the adjacent side (distance to the nearer boat). $\newline$ $\tan(43^\circ) = \frac{41 \text{ meters}}{\text{distance to nearer boat}}$ $\newline$ $\text{distance to nearer boat} = \frac{41 \text{ meters}}{\tan(43^\circ)}$
Perform calculation for nearer boat: Perform the calculation for the distance to the nearer boat. $\newline$ $\text{distance to nearer boat} = \frac{41 \text{ meters}}{\tan(43 \text{ degrees})} \approx \frac{41}{0.9325} \approx 43.98 \text{ meters}$
Calculate distance to farther boat: Calculate the distance from the lighthouse to the farther boat using the tangent function. $\newline$ The tangent of the angle to the farther boat is equal to the opposite side (height of the lighthouse) over the adjacent side (distance to the farther boat). $\newline$ $\tan(67.6^\circ) = \frac{41 \text{ meters}}{\text{distance to farther boat}}$ $\newline$ $\text{distance to farther boat} = \frac{41 \text{ meters}}{\tan(67.6^\circ)}$
Perform calculation for farther boat: Perform the calculation for the distance to the farther boat. $\newline$ $\text{distance to farther boat} = \frac{41 \text{ meters}}{\tan(67.6 \text{ degrees})} \approx \frac{41}{2.3558} \approx 17.41 \text{ meters}$
Calculate distance between boats: Calculate the distance between the two boats. The distance between the two boats is the difference between the distances from the lighthouse to each boat. distance between boats = distance to farther boat - distance to nearer boat
Perform calculation for distance between boats: Perform the calculation for the distance between the two boats. $\newline$ distance between boats = $43.98 \, \text{meters} - 17.41 \, \text{meters} \approx 26.57 \, \text{meters}$