Question

(a) How many inches will the weight in the figure rise if the pulley is rotated through an angle of 77° 40'?
(b) Through what angle, to the nearest minute, must the pulley be rotated to raise the weight 4 in.?
(a) The weight in the figure will rise ___ inches.
(Do not round until the final answer. Then round to the nearest tenth as needed.)

Answer 1

Answer:

(a) The weight in the figure will rise 11.3 inches.

(b) 24° 35'

Step-by-step explanation:

Part (a)

To find the number of inches the weight will rise if the pulley is rotated through an angle of 77° 40', we need to find the arc length of the sector of the circle with central angle 77° 40' and radius 9.32 in.

The notation 77° 40' represents an angle in degrees, minutes, and seconds. In this case, 77° is the degree part and 40' is the minute part. To convert it to degrees only, keep the degree part as it is (77°) and divide the minutes part by 60. So, 77° 40' is equivalent to:

$77^{\circ}40'=\left(77\frac{2}{3}\right)^{\circ}$

The formula for arc length in a circle, given the central angle is given in degrees is:

$\boxed{\begin{array}{l}\underline{\textsf{Arc length}}\\\\s= \pi r\left(\dfrac{\theta}{180^{\circ}}\right)\\\\\textsf{where:}\\\phantom{ww}\bullet\;\textsf{ s is the arc length.}\\\phantom{ww}\bullet\;\textsf{ r is the radius.}\\\phantom{ww}\bullet\;\textsf{ \theta is the central angle in degrees.}\end{array}}$

Substitute r = 9.32 and θ = 77²/₃° into the arc length formula and solve for s:

$s= \pi \cdot 8.32\left(\dfrac{77\frac{2}{3}^{\circ}}{180^{\circ}}\right)$

$s= 8.32\pi \left(\dfrac{233}{540}\right)$

$s= 11.2780849158...$

$s= 11.3\; \sf inches\;(nearest\;tenth)$

Therefore, the weight in the figure will rise 11.3 inches.

Part (b)

To find through what angle the pulley must be rotated to raise the weight 4 inches, we can substitute r = 9.32 and s = 4 into the arc length formula and solve for the angle θ:

$\pi \cdot 9.32\left(\dfrac{\theta}{180^{\circ}}\right)=4$

$\dfrac{\theta}{180^{\circ}}=\dfrac{4}{9.32\pi}$

$\theta=\dfrac{4\cdot 180^{\circ}}{9.32\pi}$

$\theta=24.59046331033...^{\circ}$

Finally, we need to convert the angle in degrees into degrees and minutes. To do this, keep the whole degree part (24°) and multiply the decimal part by 60 to get the minutes:

$\theta=24^{\circ} (0.59046331033...\cdot 60)'$

$\theta=24^{\circ} (35.4277986...)'$

$\theta=24^{\circ} 35'$

Therefore, the angle through which the pulley must be rotated to raise the weight 4 inches is 24° 35' to the nearest minute.