1. The radius of a spherical hot air balloon is increasing at a constant rate of 1.8 metres per minute. At what rate is the surface area of the balloon increasing when the radius is 10 metres?

2. The radius of a circular ink blot on a piece of blotting paper is increasing at a rate of 0.5 cm per second. Find the rate at which the area of the ink blot is increasing when the radius of the blot is 5 cm.

3. During a chemical reaction the temperature (°C) of a liquid is given by . (t is the time from 0 to 30 seconds). Find the maximum temperature.

4. A tour bus operator finds his profit is $(n − 20)3(70 − 3n) where n is the number of passengers. How many passengers should he take so as to get the maximum profit?

5. The velocity of a rocket is given by:

v = 1000 + (t − 20)3 − 9(t − 20) metres per second.

At what time will the rocket's velocity be a local minimum?

6. The pressure in an engine cylinder is given by:

p = 8000[1 − sin(2πt − 3)] n.m-2

At what time does this reach a maximum and what is the maximum pressure?

7. A sheet of cardboard is 90 cm by 60 cm. It is to be folded into an open box by cutting squares of side x from each corner. Find the maximum volume.

8.

An open square-based box is to contain 10 litres.

Find the dimensions of the box so as to minimise the total surface area.