Class 12th Maths Miscellaneous Exercise on Chapter 6 – Question-16

Class 12th   Class 12th Maths

16. A cylindrical tank of radius 10 m is being filled with wheat at the rate of 314 m³/h. Then the depth of wheat is increasing at what rate?

Given:

• Radius $r=10$ m

• Rate of volume increase:

  $$\frac{dV}{dt}=314{\text{ m}}^3\mathrm{/}\text{h}$$

• Let depth (height) = $h$

Volume of a cylinder:

$$V=\pi r^{2}h$$

Differentiate with respect to time $t$:

$$\frac{dV}{dt}=\pi r^2\frac{dh}{dt}$$

Substitute values:

$$314=\pi (10{)}^2\frac{dh}{dt}$$

$$314=100\pi \frac{dh}{dt}$$
$$\frac{dh}{dt}=\frac{314}{100\pi }$$

Since $\pi \approx 3.14$:

$$\frac{dh}{dt}=\frac{314}{314}=1\text{ m/h}$$

Answer:

$$\boxed{{\displaystyle 1\text{ m/h}}}$$

Correct option: (A) 1 m/h