Calculus Answers

The area, A cm3 , of a circle increases at a constant rate of 2 cm2/s . If the initial area of A is 1 cm2 , show that the radius of the circle at time t is given by r = sqrt(2t+1/pi)

The volume, V cm3 , of a cone height h is
(pi)(h^3) / 12

If h increases at a constant rate of 0.2 cm/sec and the initial height is 2 cm, express V in terms of t and find the rate of change of V at time t.

Water is poured steadily into an empty container. If the volume of water in the container after 5 seconds is 30 cm^3 , find:
a) The rate of change of volume
b) The volume of water after 12 seconds

The radius, r cm, of a circle at time t seconds is given by r = 2t^2 + 1. Express its area, A cm^2 , in terms of t and find the rate of change of the area at the instant when t=2. (Leave your answer in terms of p )

A rectangle has sides of length x cm and 2x - 4cm and the length x cm at time t seconds is given by x = 2 + 3t, (t >=0) . Show that the area, A cm^2 , the rectangle, in terms of t is A =12t + 18t^2 . Hence find the rate of change of the area at the instant when t = 2.

The volume, V cm3 , of a cube at time t seconds is given by

V = (4 + 1/3 t)^3

the rate at which its volume is increasing at the instant when t = 2.

The amount of water,V cm3 , in a leaking tank at time t seconds is given by
V = (15 - t)^3 for 0 <= t <= 15
Find the rate at which the water leaves the tank when t = 4.

The radius, r cm , of a spherical balloon at a time t seconds is given by
r = 3 + 2/(1+t)

What is the initial radius? Find the rate of change of r (w.r.t.t) when t=3

The length, l mm of an elastic string at time t seconds is given by
l = t^3/3 - 4t + 10

Find the instants (that is, the value of t ) when
a) The length is increasing at a rate of 5 mm/Sec
b) The length is decreasing at a rate of 4 mm/Sec