Advertisements
Advertisements
Question
The radius of an air bubble is increasing at the rate of 0.5 cm/sec. At what rate is the volume of the bubble increasing when the radius is 1 cm?
Advertisements
Solution
\[\text { Let r be the radius and V be the volume of the air bubble at any time t. Then },\]
\[V=\frac{4}{3}\pi r^3 \]
\[\Rightarrow\frac{dV}{dt}=4\pi r^2 \frac{dr}{dt}\]
\[\Rightarrow\frac{dV}{dt}=4\pi \left( 1 \right)^2 \times 0.5\left( \because r = 1 \text{cm and } \frac{dr}{dt} = 0 . 5 cm/\sec \right)\]
\[\Rightarrow\frac{dV}{dt} {=2\pi cm}^3 /sec\]
APPEARS IN
RELATED QUESTIONS
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, has a variable diameter `3/2 (2x + 1)` Find the rate of change of its volume with respect to x.
The 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 ______.
The volume of a sphere is increasing at the rate of 8 cm3/s. Find the rate at which its surface area is increasing when the radius of the sphere is 12 cm.
The total cost C (x) 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 ?
A stone is dropped into a quiet lake and waves move in circles at a speed of 4 cm/sec. At the instant when the radius of the circular wave is 10 cm, how fast is the enclosed area increasing?
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 ?
The top of a ladder 6 metres long is resting against a vertical wall on a level pavement, when the ladder begins to slide outwards. At the moment when the foot of the ladder is 4 metres from the wall, it is sliding away from the wall at the rate of 0.5 m/sec. How fast is the top-sliding downwards at this instance?
How far is the foot from the wall when it and the top are moving at the same rate?
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?
Sand is being poured onto a conical pile at the constant rate of 50 cm3/ minute such that the height of the cone is always one half of the radius of its base. How fast is the height of the pile increasing when the sand is 5 cm deep ?
A particle moves along the curve y = (2/3)x3 + 1. Find the points on the curve at which the y-coordinate is changing twice as fast as the x-coordinate ?
Find the point on the curve y2 = 8x for which the abscissa and ordinate change at the same rate ?
The volume of a cube is increasing at the rate of 9 cm3/sec. How fast is the surface area increasing when the length of an edge is 10 cm?
If a particle moves in a straight line such that the distance travelled in time t is given by s = t3 − 6t2+ 9t + 8. Find the initial velocity of the particle ?
The amount of pollution content added in air in a city due to x diesel vehicles is given by P(x) = 0.005x3 + 0.02x2 + 30x. Find the marginal increase in pollution content when 3 diesel vehicles are added and write which value is indicated in the above questions ?
A ladder, 5 metre long, standing on a horizontal floor, leans against a vertical wall. If the top of the ladder slides down wards at the rate of 10 cm/sec, then find the rate at which the angle between the floor and ladder is decreasing when lower end of ladder is 2 metres from the wall ?
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
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
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
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
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.
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.
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?
A particle is moving along the curve x = at2 + bt + c. If ac = b2, then particle would be moving with uniform ____________.
The rate of change of area of a circle with respect to its radius r at r = 6 cm 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?
For a circle with area \[A=\pi r^2\], what is \[\frac{dA}{dr}\]?
In calculus, which expression represents the rate of change of \(y\) with respect to \(x\) and is used in geometry, motion, business mathematics, and many real-life situations?
