A right circular cone with radius R and height H contains a liquid which evaporates at a rate proportional to its surface area in contact with air (proportionality constant k is positive). Suppose that r(t) is the radius of the liquid cone at time t. The time after which the cone is empty is

A water tank has the shape of an inverted right circular cone with its axis vertical and vertex lowermost. Its semi-vertical angle is tan^(-1)(0.5) . Water is poured into it at a constant rate of 5 cubic meter per hour. Find the rate at which the level of the water is rising at the instant when the depth of water in the tank is 4 m.

If the height of the cone is constant then find the rate of change of its curved surface area with respect to its radius.

What is the rate of change of the volume of sphere w.r.t. its surface area when its radius is 2 units ?

Water is drained from a vertical cylindrical tank by opening a valve at the base of the tank. It is known that the rate at which the water level drops is proportional to the square root of water depth y , where the constant of proportionality k >0 depends on the acceleration due to gravity and the geometry of the hole. If t is measured in minutes and k=1/(15), then the time to drain the tank if the water is 4 m deep to start with is (a) 30 min (b) 45 min (c) 60 min (d) 80 min

A uniform solid right circular cone of base radius R is joined to a uniform solid hemisphere of radius R and of the same density, as shown. The centre of mass of the composite solid lies at the centre of base of the cone. The height of the cone is

Show that the right circular cone of least curved surface and given volume has an altitude equal to sqrt(2) time the radius of the base.

A cylindrical container of radius 'R' and height 'h' is completely filled with a liquid. Two horizontal L -shaped pipes of small cross-sectional area 'a' are connected to the cylinder as shown in the figure. Now the two pipes are opened and fluid starts coming out of the pipes horizontally in opposite directions. Then the torque due to ejected liquid on the system is

A satellite P is revolving around the earth at a height = h radius of earth= (R) above equator. Another satellite Q is at a height = 2h revolving in oppisite direction. At an instant the two are at same vertical line passing through centres of sphere. Find the least time of after which again they are in this situation.

A disc of radius R is rotating with an angular omega_(0) about a horizontal axis. It is placed on a horizontal table. The coefficient of kinetic friction is mu_(k) . (a) What was the velocity of its centre of mass before being brought in contact with the table? (b) What happens to the linear velocity of a point on its rim when placed in contact with the table? (c) What happens to the linear speed of the centre of mass when disc is placed in contact with the table? (d) Which force is responsible for the effects in (b) and (c )? (e) What condition should be satisfied for rolling to being? (f) Calculate the time taken for the rolling to begin.

A circle is the locus of a point in a plane such that its distance from a fixed point in the plane is constant. Anologously, a sphere is the locus of a point in space such that its distance from a fixed point in space in constant. The fixed point is called the centre and the constant distance is called the radius of the circle/sphere. In anology with the equation of the circle |z-c|=a , the equation of a sphere of radius is |r-c|=a , where c is the position vector of the centre and r is the position vector of any point on the surface of the sphere. In Cartesian system, the equation of the sphere, with centre at (-g, -f, -h) is x^2+y^2+z^2+2gx+2fy+2hz+c=0 and its radius is sqrt(f^2+g^2+h^2-c) . Q. Equation of the sphere having centre at (3, 6, -4) and touching the plane rcdot(2hat(i)-2hat(j)-hat(k))=10 is (x-3)^2+(y-6)^2+(z+4)^2=k^2 , where k is equal to