1.  A conical vessel of height 10 ft and semi vertical angle 30⁰ is full of water. It empties in such a way that the height of the water in the vessel is decreasing at a constant rate of one inch per minute. Find the rate at which the volume of the water in the vessel is decreasing when its height is 6 feet.
Sol: Let r be the radius and h be the height of the cone.


 

2. A conical vessel whose vertical angle is 90o is placed with its axis vertical and the vertix down wards. If water flows into it at the rate of 1 c.ft. per minute, find the rate of which the level of water is rising when the height of the water in it is 2 ft.
Sol: Let 'r' be the readings and 'h' be the height of the conical vessel.
Given: vertical angle: 90^o
   semi-vertical angle:  α  =  45^o

3. a particle is moving on a straight line and the distance 's' described by the particle in time 't' is given by     Find when its velocity vanishes what is the maximum velocity?

4. Aconical vessel whose height is 1 mt. and the radius of whose base is 5 mt. is being filled with water at the rate of   cubic meter per minute. Find the rate at which the level of the water in the vessel is rising, when the depth is 4 meters.
Sol: Let 'r' be the radius and 'h' be the height of the conical vessel.
Given h  =  2r

 

5. The base radius of a cylindrical vessel full of oil is 30 cms. Oil is drawn at the rate of 27,000 c. cm per minute. Find the rate at which the level of the oil is falling in the vessel ?
Sol : Let r, h, v be the radius, height, volume of a cylindrical vessel respectively.

 

6. A balloon which is always remains spherical, is being inflated by pumping in 900 cubic centimetres of gas per second. Find the rate at which the radius of the balloon is increasing
when the radius is 15 cm.
Sol: Let r and v be the dimensions of the spherical balloon.

 

7. The surface area of a sphere increases at the rate of 20 Sq.cm per sec. At what rate does the volume of the sphere increases, when the volume is 36 π c.c ?
Sol: Let 'r' be the radius, A be the surface area, v be the volume of the sphere.


 

8. The radius of a circular disc increases at a uniform rate of 0.025 cm. per second. Find the rate at which the area of the disc increases when its radius is 15 cm.
Sol: Let r be the radius and A be the area of a circular disc.

9. A balloon is in the shape of an inverted cone surmounted by hemisphere. Diameter of the sphere is equal to the height of the cone. If h is the total height of the balloon, then how does the volume of the balloon changes with h? what is the rate of change in volume when h = 9 unit ?
Sol: Let r be the radius and v be the volume of the hemisphere.
Let r, h, v be the radius, height and volume of the cone.
Given: height of the balloon h  =  3r
and height of the cone : 2r


 

10. A man 180 cm high, walks at a uniform rate of 12 km per hour away from a lamp post of 450 cm high. Find the rate at which the length of his shadow increases.
Sol: AB is lamp post and PQ is person x = distance of the person from the lamp post
y = length of the shadow

Writer Sayyad Anwar