If the diameter of a circle is increased by 50%, what is the percentage increase
Practice Questions
Q1
If the diameter of a circle is increased by 50%, what is the percentage increase in the area of the circle? (2021)
50%
75%
100%
125%
Questions & Step-by-Step Solutions
If the diameter of a circle is increased by 50%, what is the percentage increase in the area of the circle? (2021)
Step 1: Understand that the diameter of a circle is the distance across the circle through its center.
Step 2: Know that the radius is half of the diameter.
Step 3: If the diameter increases by 50%, calculate the new diameter. For example, if the original diameter is D, the new diameter is D + 0.5D = 1.5D.
Step 4: Calculate the new radius. Since the radius is half of the diameter, the new radius is (1.5D) / 2 = 0.75D.
Step 5: The original radius is D / 2. To find the increase in radius, compare the new radius (0.75D) to the original radius (D / 2). The increase is (0.75D - D/2).
Step 6: Simplify the increase in radius. The original radius is D/2 = 0.5D, so the increase is 0.75D - 0.5D = 0.25D.
Step 7: Calculate the percentage increase in radius. The percentage increase is (increase/original) × 100% = (0.25D / 0.5D) × 100% = 50%.
Step 8: Now, calculate the area of the circle. The area A of a circle is given by the formula A = πr².
Step 9: Calculate the original area using the original radius (D/2): A_original = π(D/2)² = π(D²/4).
Step 10: Calculate the new area using the new radius (0.75D): A_new = π(0.75D)² = π(0.5625D²).
Step 11: Find the increase in area: Increase = A_new - A_original = π(0.5625D²) - π(D²/4).
Step 12: Simplify the increase in area: Increase = π(0.5625D² - 0.25D²) = π(0.3125D²).
