Question

a thread is being pulled off a spool at the rate of 57.8 cm per sec. find the radius of the spool if it makes 144 revolutions per min. (round to two decimal places as needed.) r≈□ cm

Explanation:

Step1: Convert revolutions per minute to revolutions per second

The spool makes 144 revolutions per minute. To convert to revolutions per second, divide by 60. So, the number of revolutions per second $n=\frac{144}{60}=2.4$ revolutions per second.

Step2: Find the circumference of the spool

The thread is being pulled off at a rate of $v = 57.8$ cm per second. In one - revolution, the length of the thread pulled off is equal to the circumference of the spool $C$. In $n$ revolutions per second, the length of the thread pulled off per second is $v=nC$. So, $C=\frac{v}{n}$. Substituting $v = 57.8$ cm/s and $n = 2.4$ rev/s, we get $C=\frac{57.8}{2.4}\text{ cm}$.

Step3: Use the formula for the circumference of a circle

The formula for the circumference of a circle is $C = 2\pi r$. We know $C=\frac{57.8}{2.4}\text{ cm}$, and from $C = 2\pi r$, we can solve for $r$. So, $r=\frac{C}{2\pi}=\frac{57.8}{2.4\times2\pi}$.
\[r=\frac{57.8}{4.8\pi}\approx\frac{57.8}{4.8\times3.14159}\approx\frac{57.8}{15.08}\approx3.83\text{ cm}\]

Answer:

$3.83$