1) A cylinder of radius xx is made to fit inside a sphere of radius R as shown below. (I) show that the volume of the cylinder is given by V=2πh(R²-h²)
(ii) if the radius of the sphere, R, is 12cm, find the maximum volume of the cylinder.
(iii) use the second derivative test to show that your solution gives a maximum.

2) A metal box (without a top) is to be constructed from a square sheet of metal that is 20 cm
on a side by cutting square pieces of the same size from the corners of the sheet and then
folding up the sides. Find the dimensions of the box with the largest volume that can be
constructed in this manner and prove that these values give a maximum volume.

1. A spherical balloon is deflating so that its volume is decreasing at a constant rate of 12
cm3 per hour.

Find the rate at which the surface area of the balloon is decreasing when the volume of
the balloon is 2 500cm3.
V=4/3πr³
S=4πr²
Show any derivatives that you need to find when solving this problem.

2. A cylindrical piece of wood is being turned on a lathe. If it is 20cm long and its
radius is decreasing at a rate of 2cm per minute, find the rate at which the
volume is decreasing when its diameter is 5cm.
V = πr²h