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:
[tex]77^{\circ}40'=\left(77\frac{2}{3}\right)^{\circ}[/tex]
The formula for arc length in a circle, given the central angle is given in degrees is:
[tex]\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}}[/tex]
Substitute r = 9.32 and θ = 77²/₃° into the arc length formula and solve for s:
[tex]s= \pi \cdot 8.32\left(\dfrac{77\frac{2}{3}^{\circ}}{180^{\circ}}\right)[/tex]
[tex]s= 8.32\pi \left(\dfrac{233}{540}\right)[/tex]
[tex]s= 11.2780849158...[/tex]
[tex]s= 11.3\; \sf inches\;(nearest\;tenth)[/tex]
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:
[tex]\pi \cdot 9.32\left(\dfrac{\theta}{180^{\circ}}\right)=4[/tex]
[tex]\dfrac{\theta}{180^{\circ}}=\dfrac{4}{9.32\pi}[/tex]
[tex]\theta=\dfrac{4\cdot 180^{\circ}}{9.32\pi}[/tex]
[tex]\theta=24.59046331033...^{\circ}[/tex]
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:
[tex]\theta=24^{\circ} (0.59046331033...\cdot 60)'[/tex]
[tex]\theta=24^{\circ} (35.4277986...)'[/tex]
[tex]\theta=24^{\circ} 35'[/tex]
Therefore, the angle through which the pulley must be rotated to raise the weight 4 inches is 24° 35' to the nearest minute.
