Question

The diameter of a sphere is decreased by 25%. By what percentage it’s curved surface area decrease?

Answer 1

Hint – In this question use the concept that the new diameter of the sphere will be the old diameter subtracted with 25% of the old diameter. Then use the direct basic formula for curved surface area of the sphere which is $S.A=4\pi r^2$ to obtain the % decrease in curved surface area.

Complete step-by-step answer:

As we know that the curved surface area (S.A) of the sphere is $\left(4\pi r^2\right)$ where r is the radius of the sphere as shown in figure.
$\Rightarrow S.A=4\pi r^2$ Sq. unit. ................................... (1)
Now as we know radius (r) is half of the diameter (d).
$\Rightarrow r={\displaystyle \frac{d}{2}}$ Unit.
Now substitute this value in equation (1) we have,
$\Rightarrow S.A=4\pi {\left({\displaystyle \frac{d}{2}}\right)}^2=4\pi {\displaystyle \frac{d^2}{4}}=\pi d^2$ Sq. unit.
Now it is given that the diameter of the sphere is decreased by 25%.
Therefore new diameter (d₁) = old diameter (d) – 25% of old diameter (d).
$\Rightarrow d_1=d-{\displaystyle \frac{25}{100}}d={\displaystyle \frac{3d}{4}}$ Unit.
So the new surface area (S.A)₁ of the sphere is
$\Rightarrow {\left(S.A\right)}_1=\pi {\left({\displaystyle \frac{3d}{4}}\right)}^2={\displaystyle \frac{9}{16}}\pi d^2$ Sq. unit.
So the percentage decrease (% decrease) in surface area is the ratio of difference of old surface area and new surface area to old surface area multiplied by 100.
Therefore % decrease $={\displaystyle \frac{\pi d^2-{\displaystyle \frac{9}{16}}\pi d^2}{\pi d^2}}\times 100$
Now simplify the above equation we have,
Therefore % decrease $={\displaystyle \frac{16\pi d^2-9\pi d^2}{16\pi d^2}}\times 100={\displaystyle \frac{7}{16}}\times 100=43.75$ %.
So the surface area of the sphere is decreased by 43.75%.
So this is the required answer.

Note – Let’s talk about why the question asked for a % decrease in curved surface area and not % increase when the diameter of the sphere is decreased. It’s because if diameter is decreased this means the sphere is shrinking now, and if it shrinks then eventually the curved surface area, volume and total surface area will decrease only.