61.7k views
1 vote
Find the area of the shaded regions. Give your answer as a completely simplified exact value in terms of π (no approximations).

(I got 16.3 repeating, can anyone check me on this?)

Find the area of the shaded regions. Give your answer as a completely simplified exact-example-1
User SiN
by
7.7k points

1 Answer

3 votes

Answer: (40/3)pi

This is the same as (40pi)/3

====================================

Step-by-step explanation:

A = area of the smaller circle OB

A = pi*r^2

A = pi*3^2

A = 9pi

Circle OB has an exact area of 9pi square cm

We don't want the whole circle area, since we only are focusing on a 120 degree slice. Note how 120 degrees is 120/360 = 1/3 of a full circle; therefore we want 1/3 of the area to get the area of the slice

area of slice DOB = (1/3)*(area of circle OB)

area of slice DOB = (1/3)(9pi)

area of slice DOB = 3pi

We'll use this later so let's call this M

M = 3pi

----------------

B = area of larger circle OA

B = pi*r^2

B = pi*7^2

B = 49pi

We take 1/3 of this for similar reasons as mentioned earlier. So the area of pizza slice COA is (1/3)*(49pi) = 49pi/3

We'll use this later so let N = 49pi/3

----------------

Subtract the values of N and M.

Shaded area = N - M

Shaded area = (49pi/3) - (3pi)

Shaded area = (49pi/3) - (9pi/3)

Shaded area = (49/3)pi - (9/3)pi

Shaded area = (49/3 - 9/3)pi

Shaded area = (40/3)pi

side note: I think you computed 49/3 = 16.333 repeating which was the result for the area of circle OA; however it's not the final answer

another note: leave 40/3 as a fraction and don't convert it to 13.333 repeating. This is because the fraction 40/3 is exact while 13.333 is approximate

User Ikbal
by
9.0k points

No related questions found

Welcome to QAmmunity.org, where you can ask questions and receive answers from other members of our community.

9.4m questions

12.2m answers

Categories