Math  /  Geometry

QuestionThe circle below has center OO, and its radius is 8 ft . Given that mAOB=80m \angle A O B=80^{\circ}, find the length of the arc \overparen{A D B} and the area of the shaded region. Give exact answers in terms of π\pi, and be sure to include the correct units in your answer.
Length of \overparen{A D B} : \squareπ\pi
Area of shaded region: \square

Studdy Solution

STEP 1

What is this asking? Find the length of the curved part of the circle from point AA to BB and the area of the slice of the circle that's shaded. Watch out! Don't forget to use radians when calculating arc length and area of a sector!

STEP 2

1. Convert angle to radians
2. Calculate arc length
3. Calculate area of the sector

STEP 3

First, let's **convert the angle** from degrees to radians.
We know that 180180^\circ is equivalent to π\pi radians.
So, to convert 8080^\circ to radians, we use the formula:
Radians=Degreesπ180\text{Radians} = \frac{\text{Degrees} \cdot \pi}{180}

STEP 4

Plug in the numbers:
Radians=80π180\text{Radians} = \frac{80 \cdot \pi}{180}

STEP 5

Simplify the fraction:
Radians=4π9\text{Radians} = \frac{4\pi}{9}

STEP 6

Now, let's **calculate the arc length**.
The formula for the arc length LL is:
L=rθL = r \cdot \theta
where rr is the radius and θ\theta is the angle in radians.

STEP 7

Substitute the values we know: r=8r = 8 ft and θ=4π9\theta = \frac{4\pi}{9}:
L=84π9L = 8 \cdot \frac{4\pi}{9}

STEP 8

Multiply and simplify:
L=32π9 ftL = \frac{32\pi}{9} \text{ ft}

STEP 9

Let's **calculate the area of the sector**.
The formula for the area AA of a sector is:
A=12r2θA = \frac{1}{2} \cdot r^2 \cdot \theta

STEP 10

Substitute the values we know: r=8r = 8 ft and θ=4π9\theta = \frac{4\pi}{9}:
A=12824π9A = \frac{1}{2} \cdot 8^2 \cdot \frac{4\pi}{9}

STEP 11

Calculate 82=648^2 = 64, then substitute:
A=12644π9A = \frac{1}{2} \cdot 64 \cdot \frac{4\pi}{9}

STEP 12

Multiply and simplify:
A=256π18=128π9 square feetA = \frac{256\pi}{18} = \frac{128\pi}{9} \text{ square feet}

STEP 13

Length of \overparen{A D B}: 32π9\frac{32\pi}{9} ft
Area of shaded region: 128π9\frac{128\pi}{9} square feet

Was this helpful?

Studdy solves anything!

banner

Start learning now

Download Studdy AI Tutor now. Learn with ease and get all help you need to be successful at school.

ParentsInfluencer programContactPolicyTerms
TwitterInstagramFacebookTikTokDiscord