Question

The radii of two circles are 19 cm and 9 cm respectively. Find the radius of the circle which has circumference equal to the sum of the circumferences of the two circles.

Hint:

Notice that when circumferences are added, the resulting radius is simply the sum of the individual radii. However, this is not true for areas. If areas are added (\( \pi R^2 = \pi r_1^2 + \pi r_2^2 \)), the relationship is \(R^2 = r_1^2 + r_2^2\).

Answer 1

Step 1: Understanding the Concept:
This problem involves using the formula for the circumference of a circle and setting up an equation based on the given condition.

Step 2: Key Formula or Approach:
The formula for the circumference (\(C\)) of a circle with radius (\(r\)) is: \[ C = 2\pi r \]

Step 3: Detailed Explanation:
Let the radii of the two given circles be \(r_1\) and \(r_2\).
\(r_1 = 19\) cm
\(r_2 = 9\) cm
Let their respective circumferences be \(C_1\) and \(C_2\).
\(C_1 = 2\pi r_1 = 2\pi(19) = 38\pi\) cm.
\(C_2 = 2\pi r_2 = 2\pi(9) = 18\pi\) cm.
Let the radius of the new circle be \(R\) and its circumference be \(C\).
The problem states that the circumference of the new circle is equal to the sum of the circumferences of the two smaller circles. \[ C = C_1 + C_2 \] Substitute the formulas: \[ 2\pi R = 2\pi r_1 + 2\pi r_2 \] We can factor out \(2\pi\) from the right side: \[ 2\pi R = 2\pi (r_1 + r_2) \] Divide both sides by \(2\pi\): \[ R = r_1 + r_2 \] Now, substitute the given values of the radii: \[ R = 19 + 9 = 28 \text{ cm} \]

Step 4: Final Answer:
The radius of the new circle is 28 cm.