Question

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

Answer 1

Hint: Here, we will first find the volume of the decreased diameter of the sphere. With this value of decreased diameter, we will find the value of decreased radius and hence, we will calculate the decrease in curved surface area of the sphere.

Complete step-by-step answer:
Let us take the radius of the sphere be = r.
And diameter of the sphere be = d.
We know that radius is half of the diameter, so:
\[r=\dfrac{d}{2}\]

Using, this relation we can also write:
$\Delta r=\dfrac{\Delta d}{2}$
On dividing both sides of this equation by r, we get:
$\dfrac{\Delta r}{r}=\dfrac{\Delta d}{2r}$
Since, we have 2r=d. So, we can write above equation as:
$\dfrac{\Delta r}{r}=\dfrac{\Delta d}{d}$

Now, we may multiply both sides by 100 in order to get the percentage change as:
$\dfrac{\Delta r}{r}\times 100=\dfrac{\Delta d}{d}\times 100............\left( 1 \right)$
Since, it is given that percentage change in d = 25%. So on putting the percentage change in d that is $\dfrac{\Delta d}{d}\times 100=25%$ in equation (1), we get:
$\dfrac{\Delta r}{r}\times 100=25%..........\left( 2 \right)$
So, the percentage change in radius of the sphere is also = 25%

We know that the formula for curved surface area of a sphere is given as:
$CSA=4\pi {{r}^{2}}$
Let us denote the surface area by S. So, expression for the change of surface area will be:
$\Delta S=4\pi \times 2r\times \Delta r$

On dividing both sides by $4\pi {{r}^{2}}$, we get:
$\begin{align}
  & \dfrac{\Delta S}{4\pi {{r}^{2}}}=\dfrac{4\pi \times 2r\times \Delta r}{4\pi {{r}^{2}}} \\
 & \dfrac{\Delta S}{S}=2\dfrac{\Delta r}{r} \\
\end{align}$

Now, on multiplying both sides of this equation by 100 to get percentage change, we get:
$\dfrac{\Delta S}{S}\times 100=2\dfrac{\Delta r}{r}\times 100..........(3)$
On substituting value from equation (2) in equation (3), we get:
$\dfrac{\Delta S}{S}\times 100=2\times 25%=50%$
So, the change in CSA of the sphere is 50%.
Hence, the decrease in CSA of the sphere is = 50%.

Note: Students should note here that as we have been given percentage change in this question. So, we always have to multiply the change obtained with 100 in order to get a percentage change.