Application - word problems
Task number: 3050
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.)
