Question

If the diameter of the circle below is 17 inches, find the length of the bolded arc to the nearest tenth of an inch.

a. 41.8 in
b. 44.2 in
c. 45.9 in
d. 47.3 in

Answer 1

The length of the bolded arc, excluding the 62-degree sector, is approximately 44.1 inches, making the closest option b. 44.2 in.

To find the length of the bolded arc, you need to subtract the length of the sector (made by 62 degrees) from the circumference of the entire circle.

1. Find the Circumference of the Circle:

`\[ C = \pi * \text{Diameter} \] \[ C = \pi * 17 \]`

2. Find the Length of the Sector:

The central angle of the sector is 62 degrees, which is a fraction of the entire circle. The fraction of the circumference represented by the sector is
`\( (62)/(360) \)`.

`\[ \text{Sector Length} = (62)/(360) * C \]`

3. Find the Length of the Bolded Arc:

Subtract the length of the sector from the circumference.

`\[ \text{Bolded Arc Length} = C - \text{Sector Length} \]`

Now, calculate these values and choose the answer closest to the result:

`\[ C = \pi * 17 \]\[ \text{Sector Length} = (62)/(360) * C \]\[ \text{Bolded Arc Length} = C - \text{Sector Length} \]`

After performing the calculations, choose the answer that is closest to the result among the given options.

1. Find the Circumference of the Circle:

`\[ C = \pi * \text{Diameter} \] \[ C = \pi * 17 \] \[ C \approx 53.407 \, \text{inches} \]`

2. Find the Length of the Sector:

The central angle of the sector is 62 degrees, which is a fraction of the entire circle. The fraction of the circumference represented by the sector is
`\( (62)/(360) \)`.

`\[ \text{Sector Length} = (62)/(360) * C \] \[ \text{Sector Length} \approx (62)/(360) * 53.407 \approx 9.266 \, \text{inches} \]`

3. Find the Length of the Bolded Arc:

Subtract the length of the sector from the circumference.

`\[ \text{Bolded Arc Length} = C - \text{Sector Length} \] \[ \text{Bolded Arc Length} \approx 53.407 - 9.266 \approx 44.141 \, \text{inches} \]`

Now, compare this result with the given options:

a. 41.8 in

b. 44.2 in

c. 45.9 in

d. 47.3 in

The length of the bolded arc is approximately 44.1 inches, so the closest option is b. 44.2 in.