Question

find the area of a 60 degree sector with radius 10.
select one:
a. 52.3
b. 314
c. 188
d. 209

the angle of depression of a boat at sea from a 100 foot lighthouse is 20 degrees. find the distance to the boat.
select one:
a. 292
b. 36.4
c. 106
d. 274.8

Explanation:

First Problem: Area of a Sector

Step1: Recall sector area formula

The formula for the area of a sector with central angle $\theta$ (in degrees) and radius $r$ is $A = \frac{\theta}{360^\circ} \times \pi r^2$.

Step2: Substitute given values

$\theta = 60^\circ$, $r = 10$
$A = \frac{60^\circ}{360^\circ} \times \pi \times 10^2$

Step3: Simplify the expression

$A = \frac{1}{6} \times \pi \times 100 = \frac{100\pi}{6} \approx 52.3$

Step1: Relate angle of depression

The angle of depression equals the angle of elevation from the boat to the top of the lighthouse, so $\theta = 20^\circ$. The lighthouse height ($100$ ft) is the opposite side to the angle, and the distance to the boat ($d$) is the adjacent side.

Step2: Use tangent function

$\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} = \frac{100}{d}$

Step3: Solve for $d$

$d = \frac{100}{\tan(20^\circ)}$
$\tan(20^\circ) \approx 0.3640$, so $d \approx \frac{100}{0.3640} \approx 274.8$

Answer:

a. 52.3

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Second Problem: Angle of Depression