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 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]
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?
Sand is pouring from a pipe at the rate of 12 cm3/s. The falling sand forms a cone on the ground in such a way that the height of the cone is always one-sixth of the radius of the base. How fast is the height of the sand cone increasing when the height is 4 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.
The total revenue received from the sale of x units of a product is given by R (x) = 13x2 + 26x + 15. Find the marginal revenue when x = 7 ?
The side of a square is increasing at the rate of 0.2 cm/sec. Find the rate of increase of the perimeter of the square.
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 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 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 man 160 cm tall, walks away from a source of light situated at the top of a pole 6 m high, at the rate of 1.1 m/sec. How fast is the length of his shadow increasing when he is 1 m away from the pole?
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 ladder 13 m long leans against a wall. The foot of the ladder is pulled along the ground away from the wall, at the rate of 1.5 m/sec. How fast is the angle θ between the ladder and the ground is changing when the foot of the ladder is 12 m away from the wall.
Find an angle θ which increases twice as fast as its cosine ?
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?
Water is running into an inverted cone at the rate of π cubic metres per minute. The height of the cone is 10 metres, and the radius of its base is 5 m. How fast the water level is rising when the water stands 7.5 m below the base.
A man 2 metres high walks at a uniform speed of 6 km/h away from a lamp-post 6 metres high. Find the rate at which the length of his shadow increases ?
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.
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 ?
Find the point on the curve y2 = 8x for which the abscissa and ordinate change at the same rate ?
The volume of a spherical balloon is increasing at the rate of 25 cm3/sec. Find the rate of change of its surface area at the instant when radius is 5 cm ?
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 radius of a sphere is changing at the rate of 0.1 cm/sec. The rate of change of its surface area when the radius is 200 cm is
The radius of the base of a cone is increasing at the rate of 3 cm/minute and the altitude is decreasing at the rate of 4 cm/minute. The rate of change of lateral surface when the radius = 7 cm and altitude 24 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
The volume of a sphere is increasing at the rate of 4π cm3/sec. The rate of increase of the radius when the volume is 288 π cm3, is
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
Water is dripping out at a steady rate of 1 cu cm/sec through a tiny hole at the vertex of the conical vessel, whose axis is vertical. When the slant height of water in the vessel is 4 cm, find the rate of decrease of slant height, where the vertical angle of the conical vessel is `pi/6`
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?
x and y are the sides of two squares such that y = x – x2. Find the rate of change of the area of second square with respect to the area of first square.
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 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.
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.
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?
