Advertisements
Advertisements
प्रश्न
If the rate of change of volume of a sphere is equal to the rate of change of its radius, find the radius of the sphere ?
Advertisements
उत्तर
\[\text { Let r be the radius and V be the volume of the sphere at any time t. Then },\]
\[V = \frac{4}{3}\pi r^3 \]
\[ \Rightarrow \frac{dV}{dt} = 4\pi r^2 \frac{dr}{dt}\]
\[ \Rightarrow 4\pi r^2 = 1 \left[ \because \frac{dV}{dt} = \frac{dr}{dt} \right]\]
\[ \Rightarrow r^2 = \frac{1}{4\pi}\]
\[ \Rightarrow r = \sqrt{\frac{1}{4\pi}}\]
\[ \Rightarrow r = \frac{1}{2\sqrt{\pi}} \text { units }\]
APPEARS IN
संबंधित प्रश्न
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.
A stone is dropped into a quiet lake and waves move in circles at the speed of 5 cm/s. At the instant when the radius of the circular wave is 8 cm, how fast is the enclosed area increasing?
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.
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 cost C(x) in rupees associated with the production of x units of an item is given by C(x) = 0.007x3 – 0.003x2 + 15x + 4000. Find the marginal cost when 17 units are produced
Find the rate of change of the volume of a sphere with respect to its diameter ?
Find the rate of change of the area of a circle with respect to its radius r when r = 5 cm
An edge of a variable cube is increasing at the rate of 3 cm per second. How fast is the volume of the cube increasing when the edge is 10 cm long?
A man 2 metres high walks at a uniform speed of 5 km/hr away from a lamp-post 6 metres high. Find the rate at which the length of his shadow increases.
A particle moves along the curve y = x2 + 2x. At what point(s) on the curve are the x and y coordinates of the particle changing at the same rate?
A balloon in the form of a right circular cone surmounted by a hemisphere, having a diameter equal to the height of the cone, is being inflated. How fast is its volume changing with respect to its total height h, when h = 9 cm.
The surface area of a spherical bubble is increasing at the rate of 2 cm2/s. When the radius of the bubble is 6 cm, at what rate is the volume of the bubble increasing?
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 perimeter.
The volume of a sphere is increasing at 3 cubic centimeter per second. Find the rate of increase of the radius, when the radius is 2 cms ?
The side of a square is increasing at the rate of 0.1 cm/sec. Find the rate of increase of its perimeter ?
The radius of a circle is increasing at the rate of 0.5 cm/sec. Find the rate of increase of its circumference ?
Side of an equilateral triangle expands at the rate of 2 cm/sec. The rate of increase of its area when each side is 10 cm is
A cone whose height is always equal to its diameter is increasing in volume at the rate of 40 cm3/sec. At what rate is the radius increasing when its circular base area is 1 m2?
A cylindrical vessel of radius 0.5 m is filled with oil at the rate of 0.25 π m3/minute. The rate at which the surface of the oil is rising, is
The distance moved by the particle in time t is given by x = t3 − 12t2 + 6t + 8. At the instant when its acceleration is zero, the velocity is
The coordinates of the point on the ellipse 16x2 + 9y2 = 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 3 cm3/sec. The rate at which the radius increases when radius is 2 cm, is
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
A man 2 metres tall walks away from a lamp post 5 metres height at the rate of 4.8 km/hr. The rate of increase of the length of his shadow is
In a sphere the rate of change of volume is
Two men A and B start with velocities v at the same time from the junction of two roads inclined at 45° to each other. If they travel by different roads, find the rate at which they are being seperated.
A man, 2m tall, walks at the rate of `1 2/3` m/s towards a street light which is `5 1/3`m above the ground. At what rate is the tip of his shadow moving? At what rate is the length of the shadow changing when he is `3 1/3`m from the base of the light?
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
A particle is moving along the curve x = at2 + bt + c. If ac = b2, then particle would be moving with uniform ____________.
Total revenue in rupees received from the sale of x units of a product is given by R(x)= 3x2+ 36x + 5. The marginal revenue, when x = 15 is ____________.
If the rate of change of volume of a sphere is equal to the rate of change of its radius then the surface area of a sphere is ____________.
If equal sides of an isosceles triangle with fixed base 10 cm are increasing at the rate of 4 cm/sec, how fast is the area of triangle increasing at an instant when all sides become equal?
The median of an equilateral triangle is increasing at the ratio of `2sqrt(3)` cm/s. Find the rate at which its side is increasing.
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?
Given that `1/y + 1/x = 1/12` and y decreases at a rate of 1 cms–1, find the rate of change of x when x = 5 cm and y = 1 cm.
A kite is being pulled down by a string that goes through a ring on the ground 8 meters away from the person pulling it. If the string is pulled in at 1 meter per second, how fast is the kite coming down when it is 15 meters high?
