Parthasarathi Mukhopadhyay and Manabendra Nath Mukherjee Solutions for Exercise 1: EXERCISE

The population of a country increases at the rate proportional to the number of inhabitants. If the population doubles in $30$ years, in how many years will it triple ?

[Given that, ${\log }_{e}3=1.6\left({\log }_{e}2\right)$]

A circular ink-blot grows at the rate of $2$ square cm / sec. Find the rate at which the radius is increasing after $2\frac{6}{11}$ seconds.

The time rate of change of radius of a sphere is $\frac{1}{2\pi }$ . When its radius is $5$ cm. Find the rate of change of the area of the surface of the sphere with time.

A spherical ice-ball is melting in such a way that the radius is decreasing at a constant rate of $0.1 \text{cm}/\text{sec}$. Find the amount of water formed in one second when the radius of the sphere is $7$ cm [Given $\pi =\frac{22}{7}$, specific gravity of ice $=0.9$]

Find the co-ordinates of the position of a particle moving along the parabola $y^2=4x$ at which the rate of increase of the abscissa is twice the rate of increase of the ordinate.

A solid cube changes its volume such that its shape remains unchanged. For such a cube of unit volume, show that, rate of change of volume$=\frac{3}{2}$ (rate of change of area of any face of the cube).

A light is at the top of a pole $50 \mathrm{m}$ high. A ball is dropped from the same height from a point $30 \mathrm{m}$ away from the light. How fast is the shadow of the ball moving along the ground $\frac{1}{2} \sec$  later? (Assume distance covered by ball, $s=\frac{1}{2}g{t}^2$ where, $g=9.8 \mathrm{m}/{\sec }^2$)

A light is at the top of a pole $2a$ metre high and the pole is placed at the side of a straight road. A boy $a$ metre tall is walking at a speed of $c$ metre/min towards the pole. Determine the rate of diminishing the length of his shadow.