You are making cylindrical containers to contain a given volume. Suppose that the top and bottom are made of a material that is {{MATH_14}} times as expensive (cost per unit area) as the material used for the lateral side of the cylinder. Find (in terms of {{MATH_15}}) the ratio of height to base radius of the cylinder that minimizes the cost of making the containers. {{CMD_16}} First we draw a picture: {{CMD_17}} {{ENV_18}} {{CMD_19}} Letting {{MATH_20}} represent the cost of the lateral side, we can write an expression for the cost of materials: {{MATH_21}} Since we know that {{MATH_22}}, we can use this relationship to eliminate {{MATH_23}} (we could eliminate {{MATH_24}}, but it's a little easier if we eliminate {{MATH_25}}, which appears in only one place in the above formula for cost). We find {{ENV_26}} Since we have expressed the cost as a function of a single variable, the problem becomes a``calculus problem": Find the minimum of the function {{MATH_27}} on the interval {{MATH_28}}.