Question

The volume V and depth x of water in a vessl are connected by the relation $V=5x-\frac{x^2}{6}$ and the volume of water is increasing , at the rate of $5c{m}^3/sec$, when x = 2 cm. The rate at which the depth of water is increasing, is

• A. $\frac{5}{18}cm/\sec$
• B. $\frac{1}{4}cm/\sec$
• C. $\frac{5}{16}cm/\sec$
• D. None of these

Answer 1

Answer: (D)

$V=5x-\frac{x^2}{6}\Rightarrow \frac{dV}{dt}=5\frac{dx}{dt}-\frac{x}{3}.\frac{dx}{dt}$
$\Rightarrow \frac{dx}{dt}=\frac{\frac{dV}{dt}}{\left(5-\frac{x}{3}\right)}$
$\Rightarrow {\left(\frac{dx}{dt}\right)}_{x=2}=\frac{5}{5-\frac{2}{3}}=\frac{15}{13}cm/\sec$