Solve the problem.
-A sensor light installed on the edge of a home can detect motion for a distance of $50 \mathrm { ft }$. in front and with a range of motion of $238 ^ { \circ }$. Over what area will the sensor detect motion and become illuminated? Round to the nearest hundredth.
A) $5192.35 \mathrm { ft } ^ { 2 }$
B) $10,384.81 \mathrm { ft } ^ { 2 }$
C) $10,384.71 \mathrm { ft } ^ { 2 }$
D) $5192.25 \mathrm { ft } ^ { 2 }$

Question

Quizplus · Accepted Answer

The area over which the sensor detects motion can be calculated using the formula for the area of a sector of a circle, $A = \frac{1}{2}r^2\theta$, where $r$ is the radius (in this case, 50 ft) and $\theta$ is the angle in radians. First, convert the angle from degrees to radians: $238^\circ = 238 \times \frac{\pi}{180} \approx 4.15$ radians. Then, calculate the area: $A = \frac{1}{2} \times 50^2 \times 4.15 \approx 5192.35 \, \text{ft}^2$.

Solve the Problem 50ft50 \mathrm { ft }50ft in Front and with a Range of Motion Of of a Home

Solve the Problem $50 \mathrm { ft }$ in Front and with a Range of Motion Of of a Home