Question

In the figure above, arc RST measures 30°

and the length of arc ST is 4π. What is the
area of RST?
A) 576 π
B) 192 π
C) 96 π
D) 48 π
E) 24 π

Answer 1

My brain just says this I not sure if it’s right or not but I think it’s C

Answer 2

The area of the sector RST is approximately
`\( 150.80 \, \pi \)` square units. Since the options are given in terms of multiples of
`\( \pi \),` we round to the nearest whole number, which is
`\( 151 \, \pi \)`.

To find the area of the sector RST of a circle, we can follow these steps:

1. Identify the angle of the sector. In this case, it is 30°.

2. Find the radius of the circle by using the arc length formula, which is
`\( L = \theta r \),` where L is arc length,
`\( \theta \)` is the angle in radians, and r is the radius.

3. Calculate the area of the sector, which is
`\( A = (1)/(2)r^2\theta \),`where A is the area, r is the radius, and
`\( \theta \)` is the angle in radians.

Given that the length of arc ST is
`\( 4\pi \),` and the angle of the sector is 30°, we first need to convert the angle to radians because the arc length is given in terms of pi, which suggests radians are being used:

30° =
`(\pi)/(6)`radian

Now, let's use the arc length formula
`\( L = \theta r \)` to find the radius:

`\[ 4\pi = (\pi)/(6) r \]`

Solving for r:

`\[ r = (4\pi)/((\pi)/(6)) \]`

`\[ r = 4 * 6 \]`

`\[ r = 24 \]`

Now we know the radius of the circle is 24. We can find the area of the sector using the formula
`\( A = (1)/(2)r^2\theta \):`

`\[ A = (1)/(2) * (24)^2 * (\pi)/(6) \]`

A=
`\( 150.80 \, \pi \)` square units