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

Find the volume of a right-circular cone of base radius r and height h.

Derive the formula for the volume of right circular cone with radius r and height h.

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 metre 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.

The time period of artificial satellite in a circular orbit of radius R is T. The radius of the orbit in which time period is 8T is:

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

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

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.

The surface area of a balloon being infilated changes at a constant rate. If initially, its radius 3 units and after 2 seconds it is 5 units, find the radius after t seconds.

A motorcycle has to move with a consatnt speed on an overbridge which is in the form of as circular arc of radius R and has a total length L. Suppose the motorcycle starts from the highest point. a) what can its maximum velocity be for which the contact with the road is not broken at the highest point? b) If the motorcycle goes at speed 1/sqrt2 times the maximum found in part a. where will it lose the contact with the road? c) What maximum uniform speed can it maintain on the bridge if it does not lose contact anywhere on the bridge?