Question

the blade of a circular saw rotates at a rate of 2000 revolutions per minute. what is the linear velocity in miles per hour of a point on the tip of the outer edge of a $7\frac{1}{4}$ inch diameter blade? the linear velocity is approximately $\square$ miles per hour. (round to the nearest tenth as needed.)

Explanation:

Step1: Calculate the radius

The diameter \(d = 7\frac{1}{4}=\frac{29}{4}\) inches. The radius \(r=\frac{d}{2}=\frac{29}{8}\) inches.

Step2: Convert revolutions per minute to revolutions per hour

Since there are \(60\) minutes in an hour, the number of revolutions per hour \(n = 2000\times60=120000\) revolutions per hour.

Step3: Calculate the circumference of the circle

The circumference of a circle \(C = 2\pi r\). Substituting \(r = \frac{29}{8}\) inches, we get \(C=2\pi\times\frac{29}{8}=\frac{29\pi}{4}\) inches per revolution.

Step4: Calculate the linear distance in inches per hour

The linear distance \(D\) (in inches per hour) is the number of revolutions per hour times the circumference. So \(D=n\times C=120000\times\frac{29\pi}{4}=870000\pi\) inches per hour.

Step5: Convert inches per hour to miles per hour

We know that \(1\) mile \( = 5280\times12 = 63360\) inches. Let \(v\) be the linear velocity in miles per hour. Then \(v=\frac{D}{63360}\). Substituting \(D = 870000\pi\) inches, we have \(v=\frac{870000\pi}{63360}\).

$$v=\frac{870000\times3.14159}{63360}$$

$$v=\frac{2733183.3}{63360}\approx43.1$$

Answer:

\(43.1\)