Advertisements
Advertisements
प्रश्न
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.
उत्तर
Let at ant instant of time t, the redius of the balloon be r and its volume be V, then
`V = 4/3 pir^3` ....(i)
Differentiating (i) w.r.t.t, we get,
`(dV)/dt = (4/3 pi) (3r^2 (dr)/dt)`
= 900 cm3 /sec = `(4/3 pi) {3 (15 cm)^2 (dr)/dt}` ....`(∵ (dV)/ (dr) = 900 cm^3 //sec.)`
`= (dr)/dt = 900/ (4pi xx (15)^2)` cm/sec
`1/pi` cm/sec
∴ Rate of increase of the radius of the balloon = `1/pi` 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`
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.
The Volume of cube is increasing at the rate of 9 cm 3/s. How fast is its surfacee area increasing when the length of an edge is 10 cm?
Find the rate of change of the area of a circle with respect to its radius r when r = 3 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
The rate of change of the area of a circle with respect to its radius r at r = 6 cm is ______.
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 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 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 area of a circular disc with respect to its circumference when the radius is 3 cm ?
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 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.
If y = 7x − x3 and x increases at the rate of 4 units per second, how fast is the slope of the curve changing when x = 2?
Find an angle θ whose rate of increase twice is twice the rate of decrease of its cosine ?
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.
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 radius of a circle is increasing at the rate of 0.5 cm/sec. Find the rate of increase of its circumference ?
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 ?
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
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 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
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
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
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
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
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.
If the area of a circle increases at a uniform rate, then prove that perimeter varies inversely as the radius
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 area of a circle with respect to its radius r at r = 6 cm is ____________.
If the rate of change of the area of the circle is equal to the rate of change of its diameter then its radius is equal to ____________.
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 ____________.
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?
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?
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?
