A 15,000 litre rainwater tank is to be built in the shape of a cylinder. It must be at most 3 metres tall, and have radius at most 2 metres. Its base costs 40 dollars per square metre, and the sides and top cost 20 per square metre. What are the cheapest and most expensive shapes that satisfy these conditions?
a) Give the total cost as a function of the height H and radius R. Use units of metres for both H and R. Note that it is case sensitive - you must use exactly the variable names given in the question.
b) Give an equation that connects the height H and radius R. Your answer must be an equation including "=". Remember to use consistent units.
c) Write the cost as a function just of R.
d) For which value of R does this cost function have a local minimum? Enter in metres, accurate within a centimetre.
e) What is the smallest allowed radius R, that keeps the height within bounds? Answer as a decimal in metres, accurate within a centimetre
f) What is the largest allowed radius R?
g) Enter the cheapest cost, accurate to within a dollar