Question

If the diameter of a sphere is decreased by $25\%$, then its curved surface area decrease by

• A. $44.25\%$
• B. $43.50\%$
• C. $43.75\%$
• D. $75.43\%$

Answer 1

The correct option is C

$43.75\%$

Step 1: Given data

Diameter of a sphere is decreased by $25\%$

Let the original diameter of the sphere be $2x$.

Then, original radius of the sphere $=x$

Original curved surface area $=4\pi {\mathrm{x}}^2$

Step 2: Determining the decreased diameter of the sphere

$$\begin{gathered} =2\mathrm{x}-25\%of2\mathrm{x} \\ =2\mathrm{x}-\frac{25}{100}\times 2\mathrm{x} \\ =2\mathrm{x}-0.5\mathrm{x} \\ =1.5\mathrm{x} \end{gathered}$$

Decreased radius of the sphere $=\frac{3}{4}x$

Step 3: Determining the decreased curved surface area of the sphere

$\therefore$ Decreased curved surface area will be

$$\begin{gathered} =4\pi {\left(\frac{3}{4}x\right)}^2 \\ =4\pi \times \frac{9}{16}{x}^2 \\ =\frac{9}{4}\pi x^2 \end{gathered}$$

Decrease in area

$$\begin{gathered} =4\pi x^2-\frac{9}{4}\pi x^2 \\ =\frac{16\pi x^2-9\pi x^2}{4} \\ =\frac{7}{4}\pi x^2 \end{gathered}$$

Step 4: Determining the percentage decrease in area of the sphere

Hence, percentage decrease in area will be

$$\begin{gathered} =\frac{\frac{7}{4}\pi x^2}{4\pi x^2}\times 100\% \\ =\frac{7}{16}\times 100\% \\ =\frac{175}{4}\% \\ =43.75\% \end{gathered}$$

Hence, percentage decrease in area will be $43.75\%$