I need to create a function C(r,h) for a prototype cylindrical can, given a budget of $24pi and with the top and bottom of the can made from a material costing $1 per square centimetre used.

A water storage tank is built in the shape of a cylinder with a capacity of 40,000 ft3. The cleaning costs in dollars are expressed with the following equation: C =4Trh+Tp2, In the above equation is the radius of the tank (in feet) and h is the height of the tank (in feet). The volume of the tank is constrained to 40,000 ft3. Expressed mathematically, this means V = Tr2h , So 40000= Tr2h This question has multiple parts: 1. Determine the radius of the storage tank that minimizes the cleaning cost C You may assume that the radius of the storage tank will be between 10 to 50 feet. 2. What are the minimum cleaning costs? Be sure to use calculus when you are solving this optimization problem!

You are creating a dessert called a "S’more" as follows: • You start with a (cylindrical) marshmallow with height 3 cm and radius 1 cm. • You put a marshmallow between two flat cookies, so that the flat parts of the marshmallow are against the cookies. (See picture.) • You press down on the top cookie at a constant rate of 0.5 cm/s, which flattens and compresses the marshmallow. • You may assume that the marshmallow stays a cylinder through the compression, and its volume does not change. Find the radius at the time when the rate the radius of the marshmallow is growing is the same as the rate the height of the marshmallow is shrinking. The volume V of a cylinder of radius r and height h is V = thr². h

Suppose you have a conical tank of height 4m and radius 1m filled with a liquid of density p (shown below). R= 1 H = 4 (a) Calculate the work required to pump all of the liquid to the top of the tank. (Your answer will be in terms of p.) (b) Now suppose your tank is inverted. H = 4 R = 1 Calculate the work required to pump all of the liquid to the top of the tank (now the vertex of the cone) in this case. (c) Use your answers to (a) and (b) to determine if the amount of work required to pump the water to the top of the tank depends on the orientation of the tank (i.e. whether or not the tank is inverted). If it doesn't, briefly explain why it makes sense that the work is the same. If it does, briefly explain why it makes sense that work is different.