a. Find the length of a $$ 30^{\circ} $$ arc of a circle whose radius is 8 cm . b. Explain why the arc length in part (a) is longer or shorter in centimeters if the radius had been 11 cm . c. Find the arc length when the radius is increased to 11 cm . a. The length of the $$ 30^{\circ} $$ arc is $$ \frac{30 \pi}{36} \mathrm{~cm} $$ (Simplify your answer. Type an exact answer, using $$ \pi $$ as needed. Use integers or fractions for any numbers in

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Let's solve the problem step by step.

### Part a: Find the length of a $$ 30^{\circ} $$ arc of a circle with radius 8 cm.

The formula for the arc length $$ L $$ of a circle is given by:

$$
L = \frac{\theta}{360^{\circ}} \times 2\pi r
$$

where:
- $$ \theta $$ is the angle in degrees,
- $$ r $$ is the radius of the circle.

For this part:
- $$ \theta = 30^{\circ} $$
- $$ r = 8 \, \text{cm} $$

Substituting these values into the formula:

$$
L = \frac{30}{360} \times 2\pi \times 8
$$

Now, let's simplify this expression.
Calculate the value by following steps:
- step0: Calculate:
  $$\frac{30}{360}\times 2\pi \times 8$$
- step1: Reduce the fraction:
  $$\frac{1}{12}\times 2\pi \times 8$$
- step2: Multiply the terms:
  $$\frac{4}{3}\pi $$
- step3: Multiply:
  $$\frac{4\pi }{3}$$
The length of the $$ 30^{\circ} $$ arc with a radius of 8 cm is $$ \frac{4\pi}{3} \, \text{cm} $$.

### Part b: Explain why the arc length in part (a) is longer or shorter if the radius had been 11 cm.

The arc length is directly proportional to the radius of the circle. This means that if the radius increases, the arc length will also increase, and if the radius decreases, the arc length will decrease.

In this case, since the radius is increased from 8 cm to 11 cm, the arc length will be longer when the radius is 11 cm compared to when it was 8 cm.

### Part c: Find the arc length when the radius is increased to 11 cm.

Using the same formula for arc length:

$$
L = \frac{\theta}{360^{\circ}} \times 2\pi r
$$

For this part:
- $$ \theta = 30^{\circ} $$
- $$ r = 11 \, \text{cm} $$

Substituting these values into the formula:

$$
L = \frac{30}{360} \times 2\pi \times 11
$$

Now, let's simplify this expression.
Calculate the value by following steps:
- step0: Calculate:
  $$\frac{30}{360}\times 2\pi \times 11$$
- step1: Reduce the fraction:
  $$\frac{1}{12}\times 2\pi \times 11$$
- step2: Multiply the terms:
  $$\frac{11}{6}\pi $$
- step3: Multiply:
  $$\frac{11\pi }{6}$$
The length of the $$ 30^{\circ} $$ arc with a radius of 11 cm is $$ \frac{11\pi}{6} \, \text{cm} $$.

### Summary of Results:
- Part a: The length of the $$ 30^{\circ} $$ arc with a radius of 8 cm is $$ \frac{4\pi}{3} \, \text{cm} $$.
- Part b: The arc length is longer with a radius of 11 cm because arc length is directly proportional to the radius.
- Part c: The length of the $$ 30^{\circ} $$ arc with a radius of 11 cm is $$ \frac{11\pi}{6} \, \text{cm} $$.
- **Part a:** The length of the $$ 30^{\circ} $$ arc with a radius of 8 cm is $$ \frac{4\pi}{3} \, \text{cm} $$. - **Part b:** Increasing the radius from 8 cm to 11 cm makes the arc length longer. - **Part c:** The length of the $$ 30^{\circ} $$ arc with a radius of 11 cm is $$ \frac{11\pi}{6} \, \text{cm} $$.

Upstudy AI Solution

Answer

Solution

Part a: Find the length of a arc of a circle with radius 8 cm.

Part b: Explain why the arc length in part (a) is longer or shorter if the radius had been 11 cm.

Part c: Find the arc length when the radius is increased to 11 cm.

Summary of Results:

Extra Insights

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