Question

a thread is being pulled off a spool at the rate of 70.9 cm per sec. find the radius of the spool if it makes 128 revolutions per min
r ≈ cm
(round to two decimal places as needed.)

Explanation:

Step1: Convert revolutions per minute to revolutions per second

The spool makes 128 revolutions per minute. To convert to revolutions per second, divide by 60. So the number of revolutions per second $n=\frac{128}{60}=\frac{32}{15}$ revolutions per second.

Step2: Find the circumference of the spool per - second

The linear speed $v = 70.9$ cm/s. In one - revolution, the length of the thread pulled is equal to the circumference of the spool $C = 2\pi r$. In $n$ revolutions per second, the length of the thread pulled per second is $v=n\times C=n\times2\pi r$.

Step3: Solve for the radius $r$

We know that $v = n\times2\pi r$. Rearranging the formula for $r$, we get $r=\frac{v}{2\pi n}$. Substitute $v = 70.9$ cm/s and $n=\frac{32}{15}$ into the formula:
\[

$$\begin{align*} r&=\frac{70.9}{2\pi\times\frac{32}{15}}\\ &=\frac{70.9\times15}{2\pi\times32}\\ &=\frac{1063.5}{64\pi}\\ &\approx\frac{1063.5}{64\times3.14159}\\ &=\frac{1063.5}{201.06176}\\ &\approx5.29 \end{align*}$$

\]

Answer:

$5.29$