Question

what is the perimeter of the figure below?
you may round your answer to two decimal places.
(the figure is a sector with radius 26 ft and central angle 150°)

Explanation:

Step1: Find the length of the arc

The formula for the length of an arc of a sector with radius \( r \) and central angle \( \theta \) (in degrees) is \( \text{Arc Length} = 2\pi r \times \frac{\theta}{360^\circ} \). Here, \( r = 26 \) ft and \( \theta = 360^\circ - 150^\circ= 210^\circ \) (since the arc is the major arc).
So, \( \text{Arc Length} = 2\pi \times 26 \times \frac{210}{360} \)
\( = 52\pi \times \frac{7}{12} \)
\(=\frac{364\pi}{12}=\frac{91\pi}{3}\approx\frac{91\times3.1416}{3}\approx95.29 \) ft.

Step2: Add the lengths of the two radii

The two radii are each 26 ft, so their total length is \( 26 + 26 = 52 \) ft.

Step3: Calculate the perimeter

The perimeter of the figure is the sum of the arc length and the two radii.
\( \text{Perimeter} = 95.29+ 52 = 147.29 \) ft (rounded to two decimal places). Wait, let's recalculate the arc length more accurately.

Wait, actually, the central angle for the arc: the figure is a sector with the non - arc part being a 150 - degree angle, so the arc is part of the circle with central angle \( 360 - 150=210 \) degrees. Let's recalculate the arc length:

\( \text{Arc Length}=\frac{210}{360}\times2\times\pi\times26=\frac{7}{12}\times52\pi=\frac{364\pi}{12}\approx\frac{364\times3.14159265}{12}\)

\( 364\times3.14159265 = 364\times3 + 364\times0.14159265=1092+51.5407246 = 1143.5407246\)

\( \frac{1143.5407246}{12}\approx95.295 \) ft.

Then add the two radii: \( 95.295+26 + 26=95.295 + 52 = 147.295\approx147.30 \) ft. Wait, maybe I made a mistake in the central angle. Wait, looking at the diagram, the angle given is 150 degrees, and the arc is the larger part. Wait, maybe the central angle for the arc is 210 degrees? Wait, no, maybe the figure is a sector with central angle 210 degrees? Wait, no, the diagram shows a sector where the two radii are separated by 150 degrees, but the arc is the rest of the circle. Wait, actually, the perimeter of the sector - like figure is the length of the arc plus the two radii. Wait, let's re - express the arc length formula. The formula for the length of an arc is \( s = r\theta \) (when \( \theta \) is in radians). Let's convert 210 degrees to radians: \( 210^\circ\times\frac{\pi}{180^\circ}=\frac{7\pi}{6} \) radians. Then \( s = 26\times\frac{7\pi}{6}=\frac{182\pi}{6}=\frac{91\pi}{3}\approx95.29 \) as before. Then the two radii: 26 and 26. So total perimeter: \( \frac{91\pi}{3}+26 + 26=\frac{91\pi}{3}+52 \). Let's compute this:

\( \frac{91\times3.1416}{3}+52=\frac{285.8856}{3}+52 = 95.2952+52 = 147.2952\approx147.30 \) ft.

Wait, maybe I messed up the central angle. Wait, the figure: the angle at the center is 150 degrees, so the arc is the major arc, so the central angle for the arc is \( 360 - 150 = 210 \) degrees. Yes. So the steps are:

1. Calculate arc length: \( \frac{\theta}{360}\times2\pi r \), \( \theta = 210 \), \( r = 26 \)
2. Add the two radii (each 26) to the arc length.

Answer:

\( 147.30 \) ft (or more accurately, if we calculate \( \frac{91\pi}{3}+52\approx95.295 + 52=147.295\approx147.30 \) ft)