MATH - Differential Calculus

A cubical box is to be built so that it holds 125 cu. How precisely should the edge be made so that the volume will be correct to within 3 cu cm?

A rectangular trough is 8 ft long, 2 ft across the top, and 4 ft deep. If water flows at the rate of 2 cu ft/min, how fast is the surface rising when the water is 1 ft deep?

Water is running out of a conical funnel at the rate of 1 cubic inch per sec. If the radius of the base of the funnel is 4 in and the altitude is 8 in, find the rate at which the water level is dropping when it is 2 in from the top.

Sand falls onto a conical pile at the rate of 10 cu.in/s. The radius of the base of the pile is always ½ of the altitude. How fast is the altitude of the pile increasing in in/s when it is 5 inches deep?

A stone is thrown into still water and causes concentric circular ripples. The radius of the ripples increases at the rate of 12 in/s. At what rate does the area of the ripples increases in sq. in/s when its radius is 3 inches?

A weight is attached to one end of a 33ft rope which passes over a pulley 18f above the ground. The other end is attached to a truck at a point 3ft above the ground. If the truck moves away at a rate of 2 ft/s, how fast in ft/s is the weight rising when the truck is 8ft from the spot directly under the pulley?

A conical vessel 12cm deep and with a radius of 6 cm at the top, is being filled with water. If the rate at which the water rises is 2cm/s, how fast is the volume increasing when the water is 4cm deep?

The volume of sphere is increasing at the rate of 6cc/hr. At what rate is the surface area increasing in sq.cm/hr when the radius is 50cm?

Two men A and B, start at the same point on the circumference of a circle 100m in radius. A runs at 10m/s while B runs at 8m/s in opposite direction. Find the rate at which the line distance between them is changing when the angle subtending them is 120°.

An open cylindrical through is constructed by bending a given sheet of tin and breadth 2a. Find the radius of the cylinder of which the trough forms a part when the capacity of the trough is a maximum.

Find the radius of a cylindrical measure and given a volume V so that the area of its sides and bottom shall be a minimum.

A piece of wire 36 cm long is cut in two, one part being bent in the shape of an equilateral triangle and the other in the form of a circle. Find the length of wire for the circle if the sum of the areas of these two figures is to be a minimum.

A cylindrical steam boiler is to be constructed having a capacity of 30 cu.m. The material for the side costs P25 per sq.m and for the ends P40 per sq.m. Find the radius for least cost.

The three sides of a trapezoid are each 10cm long, how long must the fourth side if the area is a maximum?

In how many equal parts can a wire 50cm long be cut so that the product of its parts is a maximum?