Question

The diameter of a sphere is decreased by $25\%$. By what percent does its curved surface area decrease?

Answer 1

Let the original diameter is $d.$

Now the diameter is decreased by $25\%$.

The new diameter $d^2$ is $d-25\%d=\left(0.75\right)d.$

The formula for curved surface area $s=\pi d^2$

Let the new surface area be $s^n=\pi (d^n{)}^2.$

Take the ratio

$s^n:s=(d^n{)}^2:d^2=(\frac{3}{4}{)}^2$

New surface area is $s^n=\left(\frac{9}{16}\right)s$

Decrease in surface area is $s-\left(\frac{9}{16}\right)s=\left(\frac{7}{16}\right)s.$

percentage decrease in surface area is $\mathrm{43.75.}$

Hence surface area decreases by $43.75\%$