Advertisements
Advertisements
Question
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.
Advertisements
Solution
Let r be the radius and V be the volume of the sphere at any time t. Then,
`V = 4/3pir^3`
`=> (dV)/(dt) = 4pir^2 (dr)/(dt)`
`=> (dr)/(dt) = 1/(4pir^2) (dV)/(dT)`
`=> (dr)/(dt) = 8/(4pi(12)^2)` [∵ r = 12 cm and `(dV)/(dt) = 8 cm^3"/sec"`]
`=> (dr)/(dt) = 1/(72pi) "cm/sec"`
Now, let S be the surface area of the sphere at any time t. Then,
S = 4πr2
`=> (dS)/(dt) = 8pir (dr)/(dt)`
`=> (dS)/(dt) = 8pi(12)xx 1/(72pi)` [∵ r = 12 cm and `(dr)/(dt) = 1/(72pi) "cm/sec"]`
`=> (dS)/(dt) = 4/3 cm^2"/sec"`
Hence, the surface area of the sphere is increasing at the rate of `4/3 cm^2"/sec"`
APPEARS IN
RELATED QUESTIONS
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?
The radius of a circle is increasing uniformly at the rate of 3 cm/s. Find the rate at which the area of the circle is increasing when the radius is 10 cm.
A stone is dropped into a quiet lake and waves move in circles at the speed of 5 cm/s. At the instant when the radius of the circular wave is 8 cm, how fast is the enclosed area increasing?
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 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 area of a circle with respect to its radius r when r = 5 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?
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?
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.
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?
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 side of a square is increasing at the rate of 0.1 cm/sec. Find the rate of increase of its perimeter ?
A cone whose height is always equal to its diameter is increasing in volume at the rate of 40 cm3/sec. At what rate is the radius increasing when its circular base area is 1 m2?
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
If s = t3 − 4t2 + 5 describes the motion of a particle, then its velocity when the acceleration vanishes, 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
Evaluate: `int (x(1+x^2))/(1+x^4)dx`
A spherical ball of salt is dissolving in water in such a manner that the rate of decrease of the volume at any instant is proportional to the surface. Prove that the radius is decreasing at a constant rate
A kite is moving horizontally at a height of 151.5 meters. If the speed of kite is 10 m/s, how fast is the string being let out; when the kite is 250 m away from the boy who is flying the kite? The height of boy is 1.5 m.
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 ____________.
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 ____________.
A spherical balloon is being inflated at the rate of 35 cc/min. The rate of increase in the surface area (in cm2/min.) of the balloon when its diameter is 14 cm, 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?
