Rate measure practice · Mixed difficulty

The rate of change of surface area of a sphere of raditis $r$ when the radius is increasing at the rate of $2 \mathrm{~cm} / \mathrm{sec}$ is proportional to

Rays from the sun are focused by a lens of diameter 5m on to a block of ice $\left( {l = 80\,kcal/10g} \right)$ and 10 g of ice is melted in 20 minutes. Calculate the heat received from the sun on $1\,c{m^2}$ per minute

A particle moves along the curve $y = {x^2} + 2x$. Then, The point on the curve such that $x\,and\,y$ coordinates of the particle change with the same rate is

A vehicle of 200 kg moving with a velocity of 6 m/sec . To stop if in $\frac{1}{{15}}\sec$ the required force in opposite direction

The rate of change of area of a square $S$ is equal to its side length. The rate of change of side of $S$ is same as that of a cube $C$. The rate of change of volume of $C$ when its side length is 2 units, is

The distance covered by a particle in $t$ second is given by $x = 3 + 8t - 4{t^2}.\;$ After 1s its velocity will be

A balloon is pumped at the rate of $\mathrm{cm}^3 /$ minute. The rate of increase of its surface area when the radius is b $\mathrm{cm}$, is -

the speed v of particle moving along a straight line is given by $a + {y^2} = {x^2}$ (where $x$ is its distance from the origin).The acceleration of the particle is

A particle is moving in a straight line. At time $t$, the distance between the particle from its starting point is given by $x = t - 6{t^2} + {t^3}$ .Its acceleration will be zero at

If the rate of change of volume of a sphere is equal to the rate of change of its radius, then its radius is equal to

The sides of an equilateral triangle are increasing at the rate of 2cm/s.The rate at . Which the area increases, when the side is 10cm is

A cone whose height is always equal to its diameter is increasing in volume at the rate of $40 \mathrm{~cm}^3 / \mathrm{sec}$. At what rate is the radius increasing when its circular base area is $1 \mathrm{~m}^2$ ?

A spherical balloon is being inflated at the rate of 30 $c{c^{\;\;}}/\mathrm{min}$. The rate of increase of the surface area of the balloon when its diameter is 14cm, is

A spherical raindrop evaporates at a rate proportional to its surface area at any instant $t$. The differential equation giving the rate of change of the radius of the raindrop is

Moving along the x-axis there are two points with $x = 10 + 6t,x = 3 + {t^2}.$ The speed with which they are reaching from each other at the time of encounter is in cm and t is in seconds)

A point moves along the curve $12y = {x^3}$ in such a way that the rate of increase of its ordinate is more than the rate of increase of abscissa. The abscissa of the point lies in the interval

A sphere of $10 \mathrm{~cm}$ radius has a uniform thickness of ice around it. If ice is melting at the rate of $50 \mathrm{~cm}^3 / \mathrm{min}$ when thickness is $5 \mathrm{~cm}$, then rate of change of thickness is

A spherical iron ball of 10 cm radius is coated with a layer of ice of uniform thickness that melts at a rate of $50\,c{m^3}/\min .$ When the thickness of ice is 5 cm, then the rate (in cm / min) at which of the thickness of ice decreases, is

A particle moves in a straight line so that $s = \sqrt t$, then its acceleration is proportional to

If a circular iron sheet of radius $30 \mathrm{~cm}$ is heated such that its area increases at the uniform rate of $6 \pi \mathrm{cm}^2 / \mathrm{hr}$, then the rate (in $\mathrm{mm} / \mathrm{hr}$ ) at which the radius of the circular sheet increases is

A solid cylinder of height $H$ has a conical portion of same height and radius $1 / 3 r d$ of height removed from it. Rain water is accumulating in it, at the rate equal to $\pi$ times the instantaneous radius of the water surface inside the hole, the time after which hole will filled with water is

The equation of motion of a particle moving along a straight line is $s = 2{t^3} - 9{t^2} + 12t,$ where the units of $s$ and $t$ are centimeter and second. The acceleration of the particle will be zero after

A particle moves along a straight line with the law of motion given by ${s^2} = a{t^2} + 2bt + c.$ then the acceleration varies are

A particle moves along a straight line according to the law $s = 16 - 2t + 3{t^3}$ , where $s$ a metre is the distance of the particle from a fixed point at the end of $t$ seconds. The acceleration of the particle at the end of $2s$ is

The points on the curve $12y = {x^3}$ whose ordinate and abscissa change at the same rate, are

The temperature of 100 g of water is to be raised from $24^\circ C$ to $90^\circ C$ by adding steam to it. Calculate the mass of the steam required for the purpose

The semi-vertical angle of a cone is $45^{\circ}$. If the height of the cone is 20.025, then its approximate lateral surface area, is

If the rate of change of area of a square plate is equal to that of the rate of change of its perimeter, then length of the side is

A stone is dropped into a quiet lake. If the waves moves in circles at the rate of 30cm/sec when the radius is 50 m, the rate of increase of enclosed area is

The edge of a cube is equal to the radius of a sphere. If the edge and the radius increase at the same rate, then the ratio of the increases in surface areas of the cube and sphere is