Question

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

None of these

Answer 2

$V=5x-\frac{x^{2}}{6} \Rightarrow \frac{d v}{d t}=5\frac{d x}{d t}\frac{x}{3}. \frac{d x}{d t}$
$\Rightarrow \frac{d x}{d t}=\frac{\frac{d v}{d t}}{\left(5-\frac{x}{3}\right)}$
$\Rightarrow\, \left(\frac{d x}{d t}\right)_{x=2} =\frac{5}{5-\frac{2}{3}}=\frac{15}{13} cm/sec .$