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 75° 40? (b) through what angle, to the nearest minute, must the pulley be rotated to raise the weight 3 in.?

Explanation:

Part (a)

Step 1: Convert angle to decimal degrees

First, we convert \( 75^\circ 40' \) to decimal degrees. Since \( 1^\circ = 60' \), we have \( 40'=\frac{40}{60}=\frac{2}{3}\approx 0.6667^\circ \). So \( 75^\circ 40' = 75+\frac{40}{60}=75.6667^\circ \). Then convert the angle to radians using the formula \( \theta_{rad}=\theta_{deg}\times\frac{\pi}{180} \). So \( \theta = 75.6667\times\frac{\pi}{180}\approx1.3206 \) radians.

Step 2: Use arc length formula

The arc length formula is \( s = r\theta \), where \( r = 9.78 \) in (radius of the pulley) and \( \theta \) is in radians. Substitute the values: \( s=9.78\times1.3206\approx12.91 \) inches.

Step 1: Use arc length formula to find angle in radians

We know that \( s = r\theta \), so \( \theta=\frac{s}{r} \). Given \( s = 3 \) in and \( r = 9.78 \) in, we have \( \theta=\frac{3}{9.78}\approx0.3067 \) radians.

Step 2: Convert radians to degrees

To convert radians to degrees, use the formula \( \theta_{deg}=\theta_{rad}\times\frac{180}{\pi} \). So \( \theta_{deg}=0.3067\times\frac{180}{\pi}\approx17.57^\circ \).

Step 3: Convert decimal degrees to minutes

The decimal part of the degree is \( 0.57^\circ \). To convert to minutes, multiply by 60: \( 0.57\times60 = 34.2'\approx34' \). So the angle is \( 17^\circ 34' \).

Answer:

\( \approx 12.91 \) inches

Part (b)