Advertisements
Advertisements
Question
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?
Advertisements
Solution
\[\text { Let r be the radius and A be the area of the circle at any time t. Then, }\]
\[A=\pi r^2 \]
\[\Rightarrow\frac{dA}{dt}=2\pi r\frac{dr}{dt}\]
\[\Rightarrow\frac{dA}{dt}=2\pi\times4\times10\left[ \because r = 4 \text { cm and } \frac{dr}{dt} = 10 cm/\sec \right]\]
\[\Rightarrow\frac{dA}{dt} {=80\pi cm}^2 /sec\]
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.
Find the rate of change of the area of a circle with respect to its radius r when r = 3 cm.
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?
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?
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 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
The sum of the perimeter of a circle and square is k, where k is some constant. Prove that the sum of their areas is least when the side of square is double the radius of the circle.
Find the rate of change of the volume of a cone with respect to the radius of its base ?
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 ?
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 ?
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?
The radius of a circle is increasing at the rate of 0.7 cm/sec. What is the rate of increase of its circumference?
The radius of a spherical soap bubble is increasing at the rate of 0.2 cm/sec. Find the rate of increase of its surface area, when the radius is 7 cm.
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?
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 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 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.
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 side of an equilateral triangle is increasing at the rate of \[\frac{1}{3}\] cm/sec. Find the rate of increase of its perimeter ?
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 ?
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 volume of a sphere is increasing at 3 cm3/sec. The rate at which the radius increases when radius is 2 cm, is
In a sphere the rate of change of surface area is
Evaluate: `int (x(1+x^2))/(1+x^4)dx`
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
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 ______.
The rate of change of volume of a sphere is equal to the rate of change of the radius than its radius equal to ____________.
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 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?
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?
