Rate measure practice · Easy

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 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

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

A pointon the parabola $y^2=18 x$ at which the ordinate increases at twice the rate of the abscissa, is

For what values of $x$ is the rate of increase of $x^3-5 x^2+5 x+8$ is twice the rate of increase of $x$ ?

If a particle moving along a line follows the law $s=\sqrt{1+t}$, then the accelertion is proportional to

A right circular cylinder which is open at the top and has a given surface area, will have the greatest volume if its height $h$ and radius $r$ are related by

The weight $W$ of a certain stock of fish is given by $W=n w$, where $n$ is the size of stock and $w$ is the average weight of a fish. If $n$ and $w$ change with time $t$ as $n=2 t^2+3$ and $w=t^2-t+2$, then the rate of change of $W$ with respect to $\mathrm{t}$ at $t=1$ is

The coordinates of the point on the ellipse $16 x^2+9 y^2=400$ where the ordinate decreases at the same rate at which the abscissa increases, are

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

If $x = - 1$ and $x = 2$ are extreme points of $f(x) = \alpha \log \left| x \right| + \beta {x^2} + x,\,$ then

The radius of a sphere is changing at the rate of $0.1 \mathrm{~cm} / \mathrm{sec}$. The rate of change of its surface area when the radius is $200 \mathrm{~cm}$ is

While measuring the side of an equilateral triangle an error of $k\%$ is made, the percentage error in its area is

If the velocity of a body moving in a straight line is proportional to the square root of the distance traversed, then it moves with

If the surface area of a sphere of radius $r$ is increasing uniformly at the rate $8 \mathrm{~cm}^2 / \mathrm{s}$, then the rate of change of its volume is :

If the rate of change of area of a circle is equal to the rate of change of its diameter, then its radius is equal to

Consider a rectangle whose length is increasing at the uniform rate of $2 \mathrm{~m} / \mathrm{sec}$, breadth is decreasing at the uniform rate of $3 \mathrm{~m} / \mathrm{sec}$ and the area is decreasing at the uniform rate of $5 \mathrm{~m}^2 / \mathrm{sec}$. If after some time the breadth of the rectangle is $2 \mathrm{~m}$ then the length of the rectangle is

A particle moves on a line according to the law $s = a{t^2} + bt + c$. If the displacement after 1 sec is 16 cm, the velocity after 2 sec is 24 cm/sec and acceleration is $8\mathrm{cm}/{\sec ^2}\mathrm{\;}$⁡ , then

If an error of $1^{\circ}$ is made in measuring the angle of a sector of radius $30 \mathrm{~cm}$, then the approximate error in its area, is

A particle moves along the X- axis as $x = u\left( {t - 2} \right) + a{\left( {t - 2} \right)^2}$

(i) The initial velocity of particle is $u$

(ii) The acceleration of particle is $a$

(iii) The acceleration of the particle is 2a

(iv) At t=2s , particle is at the origin

Among these correct statements are

If an error of $k\%$ is made in measuring the radius of a sphere, then percentage error in its volume is

Spherical rain drop evaporates at a rate proportional to its surface area. The differential equation corresponding to the rate of change of the radius of the rain drop, if the constant of proportionality is $K>0$, is

A lizard, at an initial distance of $21 \mathrm{~cm}$ behind an insect, moves from rest with an acceleration of $2 \mathrm{~cm} / \mathrm{s}^2$ and pursues the insect which is crawling uniformly along a straight line at a speed of $20 \mathrm{~cm} / \mathrm{s}$. Then the lizard will catch the insect after

If the volume of a sphere is increasing at a constant rate, then the rate at which its radius is increasing, is

If $V=\frac{4}{3} \pi r^3$, at what rate in cubic units is $V$ increasing when $r=10$ and $\frac{d r}{d t}=0.01$ ?