Question

If the radius of a spherical balloon increases by $0.1\%$ then its volume increases approximately by

• A. $0.2\%$
• B. $0.3\%$
• C. $0.4\%$
• D. $0.05\%$

Answer 1

The correct option is B $0.3\%$

$\frac{\Delta V}{V}\times 100=\frac{[\frac{4}{3}\pi {(r+\frac{r}{1000})}^3-\frac{4}{3}\pi r^3]}{\frac{4}{3}\pi r^3}\times 100$
$=[{(1+\frac{1}{1000})}^3-1]100$
$=[1+\frac{1}{{10}^9}+\frac{3}{{10}^3}+\frac{3}{{10}^6}-1]100$
$=(\frac{1}{{10}^9}+\frac{3}{{10}^3}+\frac{3}{{10}^6})100$
$=\frac{3}{10}+\frac{3}{{10}^4}+\frac{1}{{10}^7}$
$\simeq 0.3\%\text{approx}$