Advertisements
Advertisements
प्रश्न
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 ?
Advertisements
उत्तर
\[\text { Let r be the radius and V be the volume of the sphere at any time t.Then },\]
\[V = \frac{4}{3}\pi r^3 \]
\[ \Rightarrow \frac{dV}{dt} = 4\pi r^2 \frac{dr}{dt}\]
\[ \Rightarrow \frac{dr}{dt} = \frac{1}{4\pi r^2}\frac{dV}{dt}\]
\[ \Rightarrow \frac{dr}{dt} = \frac{3}{4\pi \left( 2 \right)^2} \left[ \because r = 2 \text { cm and } \frac{dV}{dt} = 3 {cm}^3 /\sec \right]\]
\[ \Rightarrow \frac{dr}{dt} = \frac{3}{16\pi} \text{cm} /\sec\]
APPEARS IN
संबंधित प्रश्न
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`
The volume of a cube is increasing at the rate of 8 cm3/s. How fast is the surface area increasing when the length of an edge is 12 cm?
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.
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 volume of a sphere is increasing at the rate of 3 cubic centimeter per second. Find the rate of increase of its surface area, when the radius is 2 cm
The total cost C(x) associated with the production of x units of an item is given by C(x) = 0.005x3 – 0.02x2 + 30x + 5000. Find the marginal cost when 3 units are produced, whereby marginal cost we mean the instantaneous rate of change of total cost at any level of output.
Find the rate of change of the volume of a sphere with respect to its diameter ?
Find the rate of change of the volume of a sphere with respect to its surface area when the radius is 2 cm ?
Find the rate of change of the volume of a cone with respect to the radius of its base ?
The money to be spent for the welfare of the employees of a firm is proportional to the rate of change of its total revenue (Marginal revenue). If the total revenue (in rupees) recieved from the sale of x units of a product is given by R(x) = 3x2 + 36x + 5, find the marginal revenue, when x = 5, and write which value does the question indicate ?
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 particle moves along the curve y = x3. Find the points on the curve at which the y-coordinate changes three times more rapidly than the x-coordinate.
Find an angle θ whose rate of increase twice is twice the rate of decrease of its cosine ?
A kite is 120 m high and 130 m of string is out. If the kite is moving away horizontally at the rate of 52 m/sec, find the rate at which the string is being paid out.
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 side of a square is increasing at the rate of 0.1 cm/sec. Find the rate of increase of its perimeter ?
If \[V = \frac{4}{3}\pi r^3\] , at what rate in cubic units is V increasing when r = 10 and \[\frac{dr}{dt} = 0 . 01\] ? _________________
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 altitude of a cone is 20 cm and its semi-vertical angle is 30°. If the semi-vertical angle is increasing at the rate of 2° per second, then the radius of the base is increasing at the rate of
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 equation of motion of a particle is s = 2t2 + sin 2t, where s is in metres and t is in seconds. The velocity of the particle when its acceleration is 2 m/sec2, is
The diameter of a circle is increasing at the rate of 1 cm/sec. When its radius is π, the rate of increase of its area is
In a sphere the rate of change of surface area is
Water is dripping out from a conical funnel of semi-vertical angle `pi/4` at the uniform rate of 2cm2/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.
The rate of change of volume of a sphere with respect to its surface area, when the radius is 2 cm, is ______.
The sides of an equilateral triangle are increasing at the rate of 2 cm/sec. The rate at which the area increases, when side is 10 cm is ______.
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 ____________.
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.
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 man 1.6 m tall walks at the rate of 0.3 m/sec away from a street light that is 4 m above the ground. At what rate is the tip of his shadow moving? At what rate is his shadow lengthening?
A particle moves along the curve 3y = ax3 + 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.
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.
