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. \( 5 \, \text{cm/sec} \)
• B. \( \frac{5}{18} \, \text{cm/sec} \)
• C. \( 1 \, \text{cm/sec} \)
• D. None of these

Hint:

Use related rates and the chain rule to find the rate of change of one variable with respect to another.

Answer 1

The Correct Option is B

To find the rate of change of the depth of water, use the chain rule to differentiate the volume equation with respect to time.
Final Answer: \[ \boxed{\frac{5}{18} \, \text{cm/sec}} \]