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9: R3: A lighthouse is east of a sailboat. The sailboat's dock is
30km south of the lighthouse. The captain measures the angle between the lighthouse and the dock and finds it to be 40 degrees. How far is the sailboat from the dock? Round to the nearest whole
km.

$9$ : R $3$ : A lighthouse is east of a sailboat. The sailboat's dock is $30 \mathrm{~km}$ south of the lighthouse. The captain measures the angle between the lighthouse and the dock and finds it to be $40$ degrees. How far is the sailboat from the dock? Round to the nearest whole $\mathrm{km}$ .

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Pythagorean theorem

Full solution

Q.

9

: R

3

: A lighthouse is east of a sailboat. The sailboat's dock is

30 \mathrm{~km}

south of the lighthouse. The captain measures the angle between the lighthouse and the dock and finds it to be

40

degrees. How far is the sailboat from the dock? Round to the nearest whole

\mathrm{km}

.

Identify Triangle Formed: Identify the triangle formed by the lighthouse, dock, and sailboat. The dock is $30\,\text{km}$ south of the lighthouse, forming a right triangle with the sailboat.
Use Tangent Function: Use the angle given, $40$ degrees, to find the distance from the sailboat to the dock using the tangent function. $\tan(40^\circ) = \frac{\text{opposite}}{\text{adjacent}} = \frac{\text{distance from sailboat to dock}}{30 \text{ km}}$ .
Solve for Distance: Rearrange the equation to solve for the distance from the sailboat to the dock. Distance $= 30 \, \text{km} \times \tan(40^\circ)$ . Calculate $\tan(40^\circ) \approx 0.8391$ . So, Distance $\approx 30 \, \text{km} \times 0.8391$ .
Perform Multiplication: Perform the multiplication to find the distance. Distance $\approx 30 \, \text{km} \times 0.8391 \approx 25.173 \, \text{km}.$
Round to Nearest km: Round the distance to the nearest whole km. Distance $\approx 25$ km.

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