Question

the minute hand of a clock extends out to the edge of the clocks face, which is a circle of radius 4 inches. what area does the minute hand sweep out between 5:05 and 5:45? round your answer to the nearest hundredth.

Explanation:

Step1: Find the central angle

The minute - hand of a clock makes a full rotation ($360^{\circ}$) in 60 minutes. From 5:05 to 5:45, the time elapsed is $45 - 5=40$ minutes. The central angle $\theta$ (in degrees) swept by the minute - hand in 40 minutes is $\theta=\frac{40}{60}\times360^{\circ}=240^{\circ}$. Convert this to radians: $\theta = 240\times\frac{\pi}{180}=\frac{4\pi}{3}$ radians.

Step2: Use the area formula for a sector of a circle

The formula for the area $A$ of a sector of a circle with radius $r$ and central angle $\theta$ (in radians) is $A=\frac{1}{2}r^{2}\theta$. Given $r = 4$ inches and $\theta=\frac{4\pi}{3}$, we substitute these values into the formula: $A=\frac{1}{2}\times4^{2}\times\frac{4\pi}{3}$.

Step3: Calculate the area

First, $4^{2}=16$. Then $\frac{1}{2}\times16\times\frac{4\pi}{3}=\frac{16\times4\pi}{6}=\frac{32\pi}{3}\approx33.51$ square inches.

Answer:

$33.51$ square inches