Question

the diameter of a sphere is decreased by 25%.by what % does it's curved surface area decrease ?

Answer 1

If we reduce the diameter by 25%, the radius will also be reduced by 25%.

let initial radius of sphere = r
initial CSA = 4πr²

after reducing it 25%,
25% of r = (25/100)×r = 0.25r
final radius = r - 0.25r = 0.75r
final CSA = 4π(0.75r²) = 4πr²×(0.5625)

decrease in CSA = initial CSA - final CSA = 4πr² -  4πr² × (0.5625)
⇒decrease in CSA = 4πr² × (1-0.5625) = 4πr² × 0.4375

% decrease in CSA = $\frac{decrease\ in\ CSA}{initial\ CSA} \times 100= \frac{0.4375 \times 4 \pi r^2}{4 \pi r^2} \times100=43.75$

Thus CSA decreases by 43.75%.

Answer 2

Let the initial radius be=r
On decreasing by 25%,new radius=r-((25/100)xr)=3r/4
Initial surface area=4πr²
Final surface area=4π(3r/4)²=4π9r/16
% of new surface area=56.25%
% of decrease in area=(100-56.25)%=43.75%