Advertisements
Advertisements
Question
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
Options
1 m/hr
0.1 m/hr
1.1 m/hr
0.5 m/hr
Advertisements
Solution
1 m/hr
\[\text { Let r, hand V be the radius, height and volume of the cylinder at any time t. Then },\]
\[V = \pi r^2 h\]
\[ \Rightarrow \frac{dV}{dt} = \pi r^2 \frac{dh}{dt}\]
\[ \Rightarrow 314 = 3 . 14 \times \left( 10 \right)^2 \frac{dh}{dt}\]
\[ \Rightarrow \frac{dh}{dt} = \frac{314}{314}\]
\[ \Rightarrow \frac{dh}{dt} = 1 m/hr\]
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]
Find the rate of change of the area of a circle with respect to its radius r when r = 3 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 (a) the perimeter, and (b) the area of the rectangle.
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.
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) 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 ______.
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 volume of a ball with respect to its radius r. How fast is the volume changing with respect to the radius when the radius is 2 cm?
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?
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 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.
The volume of metal in a hollow sphere is constant. If the inner radius is increasing at the rate of 1 cm/sec, find the rate of increase of the outer radius when the radii are 4 cm and 8 cm respectively.
A circular disc of radius 3 cm is being heated. Due to expansion, its radius increases at the rate of 0.05 cm/sec. Find the rate at which its area is increasing when radius is 3.2 cm.
The side of a square is increasing at the rate of 0.1 cm/sec. Find the rate of increase of its perimeter ?
Find the surface area of a sphere when its volume is changing at the same rate as its radius ?
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 ?
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 ?
The distance moved by the particle in time t is given by x = t3 − 12t2 + 6t + 8. At the instant when its acceleration is zero, the velocity is
The coordinates of the point on the ellipse 16x2 + 9y2 = 400 where the ordinate decreases at the same rate at which the abscissa increases, are
The radius of a sphere is increasing at the rate of 0.2 cm/sec. The rate at which the volume of the sphere increase when radius is 15 cm, is
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
Each side of an equilateral triangle is increasing at the rate of 8 cm/hr. The rate of increase of its area when side is 2 cm, 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
The diameter of a circle is increasing at the rate of 1 cm/sec. When its radius is π, the rate of increase of its area is
Evaluate: `int (x(1+x^2))/(1+x^4)dx`
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.
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 ______.
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?
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 ______.
Given that `1/y + 1/x = 1/12` and y decreases at a rate of 1 cms–1, find the rate of change of x when x = 5 cm and y = 1 cm.
