Question

Celia is staring at the clock waiting for school to end so that she can go to track practice. She notices that the 4-inch-long minute hand is rotating around the clock and marking off time like degrees on a unit circle.

Part 1: How many radians does the minute hand move from 1:25 to 1:50? (Hint: Find the number of degrees per minute first.)
Part 2: How far does the tip of the minute hand travel during that time?

You must show all of your work.

Answer 1

Jdoe0001 is right but made a typo in part 1. You are supposed to divide 360 by 60 because there are 60 minutes on a clock. That's how you would get 6 degrees per minute.

Answer 2

well, we know the minute hand is 4 inches long, thus, the radius of the circular clock is 4 then :)

part 1)

now, there are 360° in a circle.... the clock has 60 minutes, or 60 even intervals...now, how many degrees is it for each minute? well, let's divide 360° by 6 then, 360/6 is just 6. That means each minute to the next, is 6°, well, how many radians is that? well, we know there are π radians in 180°, so, how many radians in 6° then?


`\bf \begin{array}{ccllll} radians&de grees\\ \text{\textemdash\textemdash\textemdash}&\text{\textemdash\textemdash\textemdash}\\ \pi &180\\ x&6 \end{array}\implies \cfrac{\pi }{x}=\cfrac{180}{6}\implies \cfrac{\pi }{x}=30\implies \cfrac{\pi }{30}=x`

so, there are π/30 radians in 6°.

now, from 1:25 to 1:50 is 25 minutes, and thus 25 * π/30 radians, or 5π/6 radians.

part 2)

what is the arc made by those 25 minutes?


`\bf \textit{arc's length}\\\\ s=r\theta \qquad \begin{cases} r=radius\\ \theta =\textit{angle in radians}\\ ----------\\ r=4\\ \theta =(5\pi )/(6) \end{cases}\implies s=4\cdot \cfrac{5\pi }{6} \\\\\\ s=\cfrac{10\pi }{3}\implies s\approx 10.4719755\ inches`