Both circles have the same center and radius of the larger cirlce is R. if the radius of the msaller circle is 3 units less than R, whic of the following represent the area of the shaded region?
Since diagrams do not show up, I can only guess that they probably showed the shaded area as a ring or resembling a metal washer.
take area of large circle - area of small circle
= pi(R^2) - pi(R-3)^2
expand the second term, then simplify.
See if it matches one of the answers.
To find the area of the shaded region, we need to find the area of the larger circle and subtract the area of the smaller circle.
Let's start by finding the area of the larger circle. The formula for the area of a circle is A = πr^2, where A is the area and r is the radius.
Since the radius of the larger circle is R, the area of the larger circle is A_larger = πR^2.
Now, let's find the radius of the smaller circle. We are given that the radius of the smaller circle is 3 units less than R. So the radius of the smaller circle is R - 3.
Using the formula for the area of a circle, the area of the smaller circle is A_smaller = π(R - 3)^2.
To find the area of the shaded region, we subtract the area of the smaller circle from the area of the larger circle:
Shaded Area = A_larger - A_smaller
= πR^2 - π(R - 3)^2
Now, let's simplify the expression for the shaded area:
Shaded Area = πR^2 - π(R^2 - 6R + 9)
= πR^2 - πR^2 + 6πR - 9π
= 6πR - 9π
= 3π(2R - 3)
Therefore, the area of the shaded region is represented by 3π(2R - 3).