Sector Area and Gear Rotation in Trigonometry

Q1. Find the measure (in degrees) of the central angle. The area of a sector is 5π sq units.

Background

Topic: Area of a Sector of a Circle

This question tests your understanding of how to relate the area of a sector to its central angle, using the formula for sector area in terms of radius and angle.

Key formula:

$A = \frac{1}{2} r^2 \theta$

• $A$ = area of the sector

• $r$ = radius of the circle

• $\theta$ = central angle (in radians)

Step-by-Step Guidance

1. Identify the given values: $A = 5\pi$ sq units. (Check if the radius is given or needs to be inferred from the diagram.)

2. Write the sector area formula: $A = \frac{1}{2} r^2 \theta$.

3. Set up the equation with the known values and solve for $\theta$ (in radians).

4. Convert $\theta$ from radians to degrees using $\theta_{\text{deg}} = \theta_{\text{rad}} \times \frac{180}{\pi}$.

Try solving on your own before revealing the answer!

Final Answer: 72°

We solved for $\theta$ in radians and converted to degrees using the conversion factor. The central angle is 72°.

Q2. Two gears are adjusted so that the smaller gear drives the larger one. If the smaller gear rotates through an angle of 270º, through how many degrees will the larger gear rotate?

Background

Topic: Arc Length and Rotational Motion

This question tests your ability to relate the rotation of two gears based on their radii, using the concept that the arc length traveled by the edge of each gear must be equal where they touch.

Key formula:

$s = r \theta$

• $s$ = arc length

• $r$ = radius of the gear

• $\theta$ = angle (in radians)

Step-by-Step Guidance

1. Identify the radii of the two gears from the diagram: $r_1$ (smaller), $r_2$ (larger).

2. Calculate the arc length for the smaller gear: $s_1 = r_1 \theta_1$, where $\theta_1$ is 270° (convert to radians).

3. Set $s_1 = s_2$ because the arc lengths at the point of contact are equal.

4. Write $s_2 = r_2 \theta_2$ and solve for $\theta_2$ (the angle for the larger gear).

5. Convert $\theta_2$ from radians to degrees.

Try solving on your own before revealing the answer!

Final Answer: 137°

The larger gear rotates through approximately 137°, based on the ratio of the radii and the arc length equality.