Advertisements
Advertisements
प्रश्न
A balloon, which always remains spherical has a variable radius. Find the rate at which its volume is increasing with the radius when the later is 10 cm.
Advertisements
उत्तर
Let the volume of the sun = V and radius = r
`therefore V = 4/3 pir^3`
`therefore (dV)/(dr) = 4/3 pi xx 3 r^2 = 4 pir^2`
`(dV)/(dr) = 4pi xx 10 xx 10` ...[∴ r = 10 cm]
= 400 `pi` cm3/s
Thus, when the radius is 10 cm, the volume of the balloon increases at a rate of 400 π cm2/s.
APPEARS IN
संबंधित प्रश्न
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 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.
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 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 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.
Find the rate of change of the volume of a sphere with respect to its surface area when the radius is 2 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 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 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.
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 which increases twice as fast as its cosine ?
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.
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 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 the area of the rectangle.
The side of a square is increasing at the rate of 0.1 cm/sec. Find the rate of increase of its perimeter ?
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 ?
A cylindrical vessel of radius 0.5 m is filled with oil at the rate of 0.25 π m3/minute. The rate at which the surface of the oil is rising, 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 distance moved by a particle travelling in straight line in t seconds is given by s = 45t + 11t2 − t3. The time taken by the particle to come to rest 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
In a sphere the rate of change of volume is
A 13 m long ladder is leaning against a wall, touching the wall at a certain height from the ground level. The bottom of the ladder is pulled away from the wall, along the ground, at the rate of 2 m/s. How fast is the height on the wall decreasing when the foot of the ladder is 5 m away from the wall?
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?
The rate of change of area of a circle with respect to its radius r at r = 6 cm is ____________.
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 spherical balloon is filled with 4500π cubic meters of helium gas. If a leak in the balloon causes the gas to escape at the rate of 72π cubic meters per minute, then the rate (in meters per minute) at which the radius of the balloon decreases 49 minutes after the leakage began is ______.
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 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?
If a quantity \(y\) varies with another quantity \(x\) based on the rule \(y=f(x)\), which expression represents the rate of change of \(y\) with respect to \(x\)?
What does evaluating \[\left.\frac{dy}{dx}\right|_{x=x_0}\] give?
For \[A=\pi r^2\], what is the rate of change of area with respect to radius when \[r=5\] cm?
The total cost in Rupees is \[\text{C}(x)=0.005x^3-0.02x^2+30x+5000\]. Which expression is the marginal cost?
For \[\text{C}(x)=0.005x^3-0.02x^2+30x+5000\], what is the marginal cost when \[x=3\]?
What does a positive derivative mean for a quantity?
Which procedure is used in related rates?
What must be included in the final answer for a rate of change?
