The diameter of a circle is increased by 50%.The area is increased by
a. 50%
b. 100%
c. 125%
d. 200%
e. 250%
please answer and explain
old diameter: d
old area: pi/4 d^2
new diameter: 1.5d
new area: pi/4(1.5d)^2 = pi/4 d^2 (2.25)
So, new area - old area = 2.25-1 = 1.25 = 125%
To find out how the area of a circle changes when its diameter is increased by 50%, we need to know the relationship between the diameter and the area of a circle.
The formula to calculate the area of a circle is A = πr^2, where A represents the area and r represents the radius.
The diameter of a circle is twice the radius, so we can write the formula as A = π(d/2)^2, where d represents the diameter.
Now, let's consider two cases: the original diameter (d) and the increased diameter (1.5d).
Case 1: Original Circle
Area = π(d/2)^2
Case 2: Increased Diameter
Area' = π((1.5d)/2)^2
To find the percentage increase in the area, we need to calculate the ratio of the increased area to the original area and then express it as a percentage.
Ratio of Increased Area to Original Area = Area' / Area
Area' = π((1.5d)/2)^2 = π(1.5d/2)^2 = π(1.5^2 * d^2/2^2) = π * (2.25/4) * d^2 = (π * 0.5625) * d^2
Ratio of Increased Area to Original Area = ((π * 0.5625) * d^2) / (π * (d/2)^2) = 0.5625 * (d^2 / (d^2 / 4)) = 0.5625 * (4) = 2.25
To express this ratio as a percentage, we multiply by 100.
Percentage Increase = 2.25 * 100 = 225%
Therefore, the area of the circle is increased by 225%.
The correct option is e. 250%.
Since the possible options do not match the calculated value, it appears that there might be an error in the choices provided.