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. |