A circle has a radius of 5 cm . If the area is tripled, what happens to the circumference? The circumference is multiplied by 3 . The circumference is multiplied by \( \sqrt{3} \). The circumference is multiplied by \( 3 \sqrt{3} \). The circumference is multiplied by 9 . Then

To solve the problem, we need to analyze how the area and circumference of a circle are related to its radius. 1. **Known Conditions:** - Initial radius \( r = 5 \) cm. - The area of the circle is tripled. 2. **Formulas:** - The area \( A \) of a circle is given by the formula: \[ A = \pi r^2 \] - The circumference \( C \) of a circle is given by the formula: \[ C = 2\pi r \] 3. **Step 1: Calculate the initial area.** \[ A = \pi (5)^2 = 25\pi \text{ cm}^2 \] 4. **Step 2: Calculate the new area after tripling.** \[ A' = 3A = 3(25\pi) = 75\pi \text{ cm}^2 \] 5. **Step 3: Find the new radius \( r' \) that gives this new area.** We set the new area equal to the area formula: \[ 75\pi = \pi (r')^2 \] Dividing both sides by \( \pi \): \[ 75 = (r')^2 \] Taking the square root: \[ r' = \sqrt{75} = 5\sqrt{3} \text{ cm} \] 6. **Step 4: Calculate the new circumference \( C' \).** Using the new radius: \[ C' = 2\pi r' = 2\pi (5\sqrt{3}) = 10\pi\sqrt{3} \text{ cm} \] 7. **Step 5: Calculate the initial circumference \( C \).** \[ C = 2\pi (5) = 10\pi \text{ cm} \] 8. **Step 6: Determine the factor by which the circumference is multiplied.** To find the ratio of the new circumference to the original circumference: \[ \text{Factor} = \frac{C'}{C} = \frac{10\pi\sqrt{3}}{10\pi} = \sqrt{3} \] Thus, when the area of the circle is tripled, the circumference is multiplied by \( \sqrt{3} \). **Final Answer: The circumference is multiplied by \( \sqrt{3} \).**