the center of three congruent small circles are collinear, and their diameters form the diameter of the large circle, shown, whose area is 81 pi units. what is the circumference of one of the smaller circles? express your answer in simplest radical form.
let the radius of each of the smaller circles be r
so the radius of the larger circle is 3r
π(3r)^2 = 81π
9πr^2 = 81π
r^2 = 9
r=3
Circumf of smaller circle = 2πr = 6π
To find the circumference of one of the smaller circles, we can use the information given about the large circle. Let's break it down step by step:
1. We know that the area of the large circle is 81π square units. The formula for the area of a circle is A = πr^2, where A is the area and r is the radius.
Therefore, we can set up the equation 81π = π(R/2)^2, where R is the radius of the large circle.
2. Simplifying the equation, we have 81 = (R/2)^2.
Taking the square root of both sides gives us √81 = √(R/2)^2.
Simplifying further, we get 9 = R/2.
3. Now that we know the radius of the large circle (R), we need to find the radius of one of the smaller circles. Since the centers of the smaller circles are collinear, we can determine that the diameter of the large circle is equal to the sum of the diameters of the three smaller circles.
The diameter of the large circle is equal to 2R, and the diameter of one of the smaller circles is equal to 2r, where r is the radius of the small circle.
4. From step 3, we have the equation 2R = 2r + 2r + 2r, since there are three smaller circles.
Simplifying, we get 2R = 6r.
5. Substituting the value of R we found in step 2 (R = 18), we have 2(18) = 6r.
Simplifying further, we get 36 = 6r.
6. Dividing both sides of the equation by 6 gives us r = 6.
7. Now that we know the radius of one of the smaller circles (r = 6), we can find the circumference of the circle using the formula C = 2πr.
Substituting the value of r, we get C = 2π(6) = 12π units.
Therefore, the circumference of one of the smaller circles is 12π units.