Question

refer to the figure to the right.
(a) how many inches will the weight in the figure rise if the pulley is rotated through an angle of 72° 50?
(b) through what angle, to the nearest minute, must the pulley be rotated to raise the weight 5 in.?
(a) the weight in the figure will rise
(do not round until the final answer. then round to the nearest tenth as needed.)
(b) in order for the weight to rise 5 inches, the pulley must rotate through
(do not round until the final answer. then round to the nearest minute as needed.)

Explanation:

Step1: Convert angle to radians

First, convert $72^{\circ}50'$ to decimal - degrees. $50'=\frac{50}{60}\approx0.8333^{\circ}$, so $72^{\circ}50'\approx72.8333^{\circ}$. Then convert to radians: $\theta = 72.8333\times\frac{\pi}{180}\approx1.271$ radians. The radius of the pulley $r = 9.17$ inches. The arc - length formula is $s=r\theta$.

Step2: Calculate the rise of the weight for part (a)

Substitute $r = 9.17$ inches and $\theta\approx1.271$ radians into the arc - length formula $s=r\theta$. So $s=9.17\times1.271\approx11.7$ inches.

Step3: Use the arc - length formula for part (b)

We know $s = 5$ inches and $r = 9.17$ inches. From $s=r\theta$, we can solve for $\theta$: $\theta=\frac{s}{r}=\frac{5}{9.17}\approx0.545$ radians.

Step4: Convert radians to degrees and minutes for part (b)

First, convert radians to degrees: $\theta_d=\theta\times\frac{180}{\pi}=0.545\times\frac{180}{\pi}\approx31.2^{\circ}$. Then convert the decimal part of the degrees to minutes. $0.2^{\circ}\times60 = 12'$. So $\theta\approx31^{\circ}12'$.

Answer:

(a) $11.7$
(b) $31^{\circ}12'$