Given:
In a cone, the radius is increased by 20% and height is reduced by 20%
Concept used:
Overall percentage change in volume is the successive percentage change in radius and height
Formula used:
Successive percent of x% and y% = {x + y + (xy/100)}%
Calculation:
Radius is increased by 20% and height id decreased by 20%
In the formula of volume of the cone, we have raduis2
So, here we calculate the successive percentage of 20%, 20% and -20%
The successive % of 20% and 20% = {20 + 20 + (20 × 20/100)}%
⇒ 44%
Now, the successive % of 44% and -20% = {44 - 20 - (44 × 20/100)}%
⇒ (24 - 8.8)% = 15.2%
∴ The percentage increase in the volume of the figure is 15.2%.
Traditional Method:
Given:
In a cone, the radius is increased by 20% and height is reduced by 20%
Formula used:
The volume of Cone = (1/3) × πr2h
Where r and h are the radius and height of the cone respectively
Calculation:
Let r and h be the radius and height of the cone respectively
Then, according to the formula, we have
The volume of the cone = (1/3) × πr2h
Now, r is increased by 20%
So increased radius = r + 20% of r = r + (r/5) = 6r/5
And, height is decreased by 20%
So, decreased height = h - 20% of h = h - (h/5) = 4h/5
Then, the new volume of cone = (1/3) × π(6r/5)2(4h/5) = (1/3) × πr2h(144/125)
Now, % change in the volume = [{(1/3) × πr2h(144/125) - (1/3) × πr2h}/{(1/3) × πr2h}] × 100
⇒ {(144/125) - 1} × 100 = (19/125) × 100
⇒ 0.152 × 100 = 15.2
∴ The percentage increase in the volume of the figure is 15.2%.