## Application - word problems

Solve:

• #### Variant 1

Which rectangle with perimeter $$l$$ has the greatest volume?

• #### Variant 2

Which cylinder with volume $$V$$ has the smallest surface area?

• #### Variant 3

We cut out small squares from the corners of a square sheet of paper and fold it to make a box (without a lid). How large should the cut-out squares be to maximize the volume of the box that is formed?

• #### Variant 4

How large a snowman (from three snowballs) is it possible to make from a snowball of radius 1 meter? Hint: Use Jensen's inequality. For a convex function $$f$$ and numbers $$\alpha_i$$, $$x_i$$ such that $$\alpha_i \ge 0$$, $$\sum_i \alpha_i = 1$$, $f(\sum_i \alpha_i x_i) \le \sum_i \alpha_i f(x_i) \,.$

• #### Variant 5

A corridor of width $$B$$ turns off from a corridor of width $$A$$. What is the length of the longest rod that it is possible to carry around the intersection? (For simplicity, assume that we want to carry the rod horizontally.)