U3P2 - Optimization Review

A closed cylindrical aluminum can has a volume of $$16\pi\ in^3$$ . Find the dimensions of the can that minimizes the amount of aluminum used.

$$v=\left(Area\ of\ base\right)\times h$$

$$SA=2\left(Area\ of\ base\right)+\left(Circumference\right)\left(h\right)$$

A packing company is creating a rectangular prism box with a square base. It will need to have a volume of $$864\ cm^3$$ . Find the dimensions of the prism that minimizes the amount of cardboard needed.

$$V=\left(Area\ of\ base\right)\times h$$

$$SA=\left(Area\ of\ base\right)+4bh$$

A supermarket employee wants to construct an open-top box from a 14 by 30 inch piece of cardboard. To do this, the employee plans to cut out squares of equal size from the four corners so the four sides can be bent upwards to create the box. What size should the squares be in order to create a box with the largest possible volume?

An open topped box is being constructed from a 50 by 20 inch piece of cardboard by cutting squares out of equal size from the four corners so the four sides can be bent upwards to create the box. What size should the squares be in order to create a box with the largest possible volume?

A cylindrical can holds $$1000cm^3$$ of liquid. How should the height and radius be chosen to minimize the amount of material needed to manufacture the can?

$$V=\pi r^2h$$

$$SA=2\pi r^2+2\pi rh$$