Question

the minute hand of a clock extends out to the edge of the clocks face, which is a circle of radius 2 inches. what area does the minute hand sweep out between 7:15 and 7:25? round your answer to the nearest hundredth.

Explanation:

Step1: Find the central angle

The minute - hand of a clock moves 360° in 60 minutes. From 7:15 to 7:25, it moves for 10 minutes. So the central angle $\theta$ is $\frac{10}{60}\times360^{\circ}=60^{\circ}=\frac{\pi}{3}$ radians.

Step2: Use the formula for the area of a sector

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

Step3: Calculate the area

$A=\frac{1}{2}\times4\times\frac{\pi}{3}=\frac{2\pi}{3}\approx2.09$ square inches.

Answer:

$2.09$ square inches