Question

Circle P is inside Circle Q, and the two circles share the same center X. If the circumference of Q is four times the circumference of P, and the radius of Circle P is three, what is the difference between Circle Q's diameter and Circle P's diameter?

• A. 6
• B. 9
• C. 12
• D. 18
• E. 24

Hint:

For concentric circles, the ratio of their circumferences is the same as the ratio of their radii and the ratio of their diameters. If \(C_Q/C_P = 4\), then \(r_Q/r_P = 4\) and \(d_Q/d_P = 4\). You can use this shortcut to find the diameter of Q (\(d_Q = 4 \times d_P\)) and then compute the difference.

Answer 1

The Correct Option is D

Step 1: Understanding the Concept:
The problem involves the relationship between the circumference, radius, and diameter of two concentric circles. The circumference of a circle is directly proportional to its radius and its diameter.

Step 2: Key Formula or Approach:
The key formulas for a circle are: \[\begin{array}{rl} \bullet & \text{Circumference \( C = 2 \pi r \), where \(r\) is the radius.} \\ \bullet & \text{Diameter \( d = 2r \).} \\ \end{array}\] From these, we can see that \( C = \pi d \). This means that if the circumference is scaled by a factor, the diameter and radius are scaled by the same factor.

Step 3: Detailed Explanation:
Let \(C_P, r_P, d_P\) be the circumference, radius, and diameter of Circle P.
Let \(C_Q, r_Q, d_Q\) be the circumference, radius, and diameter of Circle Q.
We are given: \begin{enumerate} \item \(C_Q = 4 \times C_P\) \item \(r_P = 3\) \end{enumerate} Since \(C = 2 \pi r\), the first condition can be written as: \[ 2 \pi r_Q = 4 \times (2 \pi r_P) \] We can cancel \(2\pi\) from both sides: \[ r_Q = 4 \times r_P \] This shows that the radius of Circle Q is also four times the radius of Circle P.
Now, we can find the radius of Circle Q: \[ r_Q = 4 \times 3 = 12 \] Next, we calculate the diameters of both circles: \[ d_P = 2 \times r_P = 2 \times 3 = 6 \] \[ d_Q = 2 \times r_Q = 2 \times 12 = 24 \] Finally, we find the difference between their diameters: \[ \text{Difference} = d_Q - d_P = 24 - 6 = 18 \] Step 4: Final Answer
The difference between Circle Q's diameter and Circle P's diameter is 18.