📘 calculus optimization

1. **State the problem:** We need to find the dimensions of a box with a square base and an open top that has a volume of 4 ft³ and requires the least material, meaning the surface

1. **State the problem:** We need to find the dimensions (radius $r$ and height $h$) of a cylindrical can that holds a volume of 1500 cm³ and minimizes the surface area (material u

1. **State the problem:** A manufacturer wants to design a cylindrical can that holds a volume of 1500 cm³. We need to find the radius $r$ and height $h$ that minimize the surface

1. **Problem statement:** A 12 cm wire is cut into two parts. One part forms a square, the other forms a circle. We want to find the radius of the circle that minimizes the sum of

1. **Problem Statement:** We need to design a square-based, open-top tank with volume 500 ft³ that minimizes the weight of the steel used. 2. **Given:**

1. **State the problem:** We want to find the radius $r$ and height $h$ of an open-top cylindrical can that holds volume $500$ cm³ and minimizes the surface area (material used). 2

1. **State the problem:** A cone-shaped cup must hold a volume of 100 cm³. We want to find the height $h$ and radius $r$ of the cone that minimize the surface area, which correspon