###### Advertisements

###### Advertisements

An edge of a variable cube is increasing at the rate of 3 cm/s. How fast is the volume of the cube increasing when the edge is 10 cm long?

###### Advertisements

#### Solution

Let the length of the cube = x cm

= Volume of the cube, V = x^{3}

Given, `dx/dt = 3` cm/s

= Rate of change of V with respect to t, `(dV)/dt = 3x^2 dx/dt`

`= (dV)/dt = 3x^2 (3)`

`= (dV)/dt = 9x^2`

`= 9 xx (10)^2 `

= 900 cm^{3}/sec

=> Rate of increase of volume of cube is 900 cm^{3}/sec

#### APPEARS IN

#### RELATED QUESTIONS

If y = f (u) is a differential function of u and u = g(x) is a differential function of x, then prove that y = f [g(x)] is a differential function of x and `dy/dx=dy/(du) xx (du)/dx`

A point source of light is hung 30 feet directly above a straight horizontal path on which a man of 6 feet in height is walking. How fast will the man’s shadow lengthen and how fast will the tip of shadow move when he is walking away from the light at the rate of 100 ft/min.

The rate of growth of bacteria is proportional to the number present. If, initially, there were

1000 bacteria and the number doubles in one hour, find the number of bacteria after 2½

hours.

[Take `sqrt2` = 1.414]

The length *x* of a rectangle is decreasing at the rate of 5 cm/minute and the width *y* is increasing at the rate of 4 cm/minute. When *x* = 8 cm and *y* = 6 cm, find the rates of change of (a) the perimeter, and (b) the area of the rectangle.

A balloon, which always remains spherical on inflation, is being inflated by pumping in 900 cubic centimetres of gas per second. Find the rate at which the radius of the balloon increases when the radius is 15 cm.

A ladder 5 m long is leaning against a wall. The bottom of the ladder is pulled along the ground, away from the wall, at the rate of 2 cm/s. How fast is its height on the wall decreasing when the foot of the ladder is 4 m away from the wall?

The radius of an air bubble is increasing at the rate `1/2` cm/s. At what rate is the volume of the bubble increasing when the radius is 1 cm?

The total revenue in rupees received from the sale of x units of a product is given by R(x) = 13x^{2} + 26x + 15. Find the marginal revenue when x = 7.

Find the rate of change of the area of a circular disc with respect to its circumference when the radius is 3 cm ?

Find the rate of change of the volume of a cone with respect to the radius of its base ?

Find the rate of change of the area of a circle with respect to its radius r when r = 5 cm

Find the rate of change of the volume of a ball with respect to its radius *r*. How fast is the volume changing with respect to the radius when the radius is 2 cm?

The total revenue received from the sale of *x* units of a product is given by R (x) = 13x^{2} + 26x + 15. Find the marginal revenue when x = 7 ?

The radius of a circle is increasing at the rate of 0.7 cm/sec. What is the rate of increase of its circumference?

A balloon which always remains spherical, is being inflated by pumping in 900 cubic centimetres of gas per second. Find the rate at which the radius of the balloon is increasing when the radius is 15 cm.

A man 180 cm tall walks at a rate of 2 m/sec. away, from a source of light that is 9 m above the ground. How fast is the length of his shadow increasing when he is 3 m away from the base of light?

A particle moves along the curve y = x^{2} + 2x. At what point(s) on the curve are the x and y coordinates of the particle changing at the same rate?

If y = 7x − x^{3} and x increases at the rate of 4 units per second, how fast is the slope of the curve changing when x = 2?

A particle moves along the curve y = x^{3}. Find the points on the curve at which the y-coordinate changes three times more rapidly than the x-coordinate.

Find an angle *θ *which increases twice as fast as its cosine ?

The radius of a cylinder is increasing at the rate 2 cm/sec. and its altitude is decreasing at the rate of 3 cm/sec. Find the rate of change of volume when radius is 3 cm and altitude 5 cm.

A particle moves along the curve *y* = (2/3)*x*^{3} + 1. Find the points on the curve at which the *y*-coordinate is changing twice as fast as the *x*-coordinate ?

The length x of a rectangle is decreasing at the rate of 5 cm/minute and the width y is increasing at the rate of 4 cm/minute. When x = 8 cm and y = 6 cm, find the rates of change of the area of the rectangle.

A circular disc of radius 3 cm is being heated. Due to expansion, its radius increases at the rate of 0.05 cm/sec. Find the rate at which its area is increasing when radius is 3.2 cm.

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

The radius of a sphere is increasing at the rate of 0.2 cm/sec. The rate at which the volume of the sphere increase when radius is 15 cm, is

The volume of a sphere is increasing at the rate of 4π cm^{3}/sec. The rate of increase of the radius when the volume is 288 π cm^{3}, is

The equation of motion of a particle is *s* = 2*t*^{2} + sin 2*t*, where *s* is in metres and *t *is in seconds. The velocity of the particle when its acceleration is 2 m/sec^{2}, is

The radius of a circular plate is increasing at the rate of 0.01 cm/sec. The rate of increase of its area when the radius is 12 cm, is

A cylindrical tank of radius 10 m is being filled with wheat at the rate of 314 cubic metre per hour. Then the depth of the wheat is increasing at the rate of

Find the rate of change of the area of a circle with respect to its radius r when r = 4 cm.

A 13 m long ladder is leaning against a wall, touching the wall at a certain height from the ground level. The bottom of the ladder is pulled away from the wall, along the ground, at the rate of 2 m/s. How fast is the height on the wall decreasing when the foot of the ladder is 5 m away from the wall?

Evaluate: `int (x(1+x^2))/(1+x^4)dx`

Water is dripping out from a conical funnel of semi-vertical angle `pi/4` at the uniform rate of 2cm^{2}/sec in the surface area, through a tiny hole at the vertex of the bottom. When the slant height of cone is 4 cm, find the rate of decrease of the slant height of water.

A spherical ball of salt is dissolving in water in such a manner that the rate of decrease of the volume at any instant is proportional to the surface. Prove that the radius is decreasing at a constant rate

If the area of a circle increases at a uniform rate, then prove that perimeter varies inversely as the radius

The volume of a cube increases at a constant rate. Prove that the increase in its surface area varies inversely as the length of the side

x and y are the sides of two squares such that y = x – x^{2}. Find the rate of change of the area of second square with respect to the area of first square.

A ladder, 5 meter long, standing on a horizontal floor, leans against a vertical wall. If the top of the ladder slides downwards at the rate of 10 cm/sec, then the rate at which the angle between the floor and the ladder is decreasing when lower end of ladder is 2 metres from the wall is ______.

#### If the rate of change of the area of the circle is equal to the rate of change of its diameter then its radius is equal to ____________.

#### The rate of change of volume of a sphere is equal to the rate of change of the radius than its radius equal to ____________.

Let y = f(x) be a function. If the change in one quantity 'y’ varies with another quantity x, then which of the following denote the rate of change of y with respect to x.

What is the rate of change of the area of a circle with respect to its radius when, r = 3 cm

The radius of a circle is increasing uniformly at the rate of 3 cm per second. Find the rate at which the area of the circle is increasing when the radius is 10 cm.

A ladder 5 m long is leaning against a wall. The bottom of the ladder is pulled along the ground, away from the wall, at the rate of 2 cm/s. How fast is its height on the wall decreasing when the foot of the ladder is 4 m away from the wall?

A cylindrical tank of radius 10 feet is being filled with wheat at the rate of 3/4 cubic feet per minute. The then depth of the wheat is increasing at the rate of

A spherical balloon is being inflated at the rate of 35 cc/min. The rate of increase in the surface area (in cm^{2}/min.) of the balloon when its diameter is 14 cm, is ______.

A spherical balloon is filled with 4500π cubic meters of helium gas. If a leak in the balloon causes the gas to escape at the rate of 72π cubic meters per minute, then the rate (in meters per minute) at which the radius of the balloon decreases 49 minutes after the leakage began is ______.

A particle moves along the curve 3y = ax^{3} + 1 such that at a point with x-coordinate 1, y-coordinate is changing twice as fast at x-coordinate. Find the value of a.

If the circumference of circle is increasing at the constant rate, prove that rate of change of area of circle is directly proportional to its radius.

An edge of a variable cube is increasing at the rate of 10 cm/sec. How fast will the volume of the cube increase if the edge is 5 cm long?