11. The minute hand of a clock is 8 inch long. How far (to the nearest tenth) does the tip of the
minute hand move in 45 minutes?
per hour
12. The diameter of each wheel of a bicycle is 30 inches. If you are traveling at 38 miles
on this bicycle, through how many revolution per minute are the wheels turning? Answer to
the nearest whole number.
13. Calculate the area of the shaded region in the given circle to the nearest hundredth, if the
radius is r = 10in.
072°
B
A

To find how far the tip of the minute hand moves in 45 minutes, given that the minute hand is 8 inches long, we can use the formula for the arc length of a circle, where the radius (r) is the length of the minute hand.

[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]

A clock completes a full circle in 60 minutes, so in 45 minutes, the central angle θ can be calculated as:

[tex]\theta=\dfrac{45}{60} \cdot 360^{\circ}[/tex]

[tex]\theta=270^{\circ}[/tex]

Now, substitute the values into the formula:

[tex]s= \pi (8)\left(\dfrac{270^{\circ}}{180^{\circ}}\right)[/tex]

[tex]s= \pi (8)\left(\dfrac{3}{2}}\right)[/tex]

[tex]s= 12\pi[/tex]

[tex]s=37.7\; \sf inches\;(nearest\;tenth)[/tex]

Therefore, the tip of the minute hand moves 37.7 inches in 45 minutes.

[tex]\hrulefill[/tex]

Question 12

To find the number of revolutions per minute (RPM) for the bicycle wheels, we need to divide the speed by the circumference of the wheel.

First find the circumference of the wheel using the formula C = πd, where d is the diameter.

Given that the diameter of the wheels is 30 inches:

[tex]C = 30 \pi \;\textsf{inches}[/tex]

Now, convert the speed from miles per hour to inches per minute.

Since 1 mile = 63360 inches and 1 hour = 60 minutes, then:

[tex]\sf \dfrac{38\;miles}{1\;hour} \times \dfrac{63360\;inches}{1\;mile}\times \dfrac{1\;hour}{60\;minutes}=40128\; in/min[/tex]

So the speed is 40,128 inches per minute.

Now, calculate the number of revolutions per minute by dividing the speed by the circumference:

[tex]\textsf{RPM} = \dfrac{\textsf{Speed}}{\textsf{Circumference}}[/tex]

[tex]\textsf{RPM} = \dfrac{40128}{30\pi}[/tex]

[tex]\textsf{RPM} = 426\; \textsf{nearest whole number}[/tex]

Therefore, the wheels are turning at 426 revolutions per minute.

[tex]\hrulefill[/tex]

Question 13

To calculate the area of the shaded region in the given circle, we can use the formula for the area of a sector of a circle:

[tex]\boxed{\begin{array}{l}\underline{\textsf{Area of a Sector}}\\\\A= \left(\dfrac{\theta}{360^{\circ}}\right) \pi r^2\\\\\textsf{where:}\\\phantom{ww}\bullet\;\textsf{$A$ is the area.}\\\phantom{ww}\bullet\;\textsf{$r$ is the radius.}\\\phantom{ww}\bullet\;\textsf{$\theta$ is the central angle in degrees.}\end{array}}[/tex]

In this case:

r = 10 in
θ = 360° - 72° = 288°

Substitute the values into the formula and solve for A:

[tex]A= \left(\dfrac{288^{\circ}}{360^{\circ}}\right) \pi \cdot 10^2[/tex]

[tex]A= \left(\dfrac{8}{10}\right) \pi \cdot 100[/tex]

[tex]A= \left(\dfrac{800}{10}\right) \pi[/tex]

[tex]A= 80 \pi[/tex]

[tex]A= 251.33\; \sf in^2\;(nearest\;hundredth)[/tex]

Therefore, the area of the shaded region is 251.33 in².