1. The base of a cuboid with height 5 is a rectangle with a perimeter of 4.

a) Determine the lengths of the sides of the base so that the cuboid has a maximum volume.

b) Express the volume of the cuboid as a function of one of the base sides. Determine the domain of this function.

c) Draw the graph of the function obtained in b).

Given one base side \(a=2\) and height \(h=5\) of the cuboid. How does the volume now depend on the perimeter of the base? What are the permissible values for this volume? Show the corresponding function graphically.
Solution. Let the base edges of the cuboid be $a$ and $b$. Then $a+b=2$, so for the volume of the cuboid we get

$$
V=5 a b=5 a(2-a)=10-5 a^{2}
$$

where $04$ m. The graph of this function is shown in the diagram on the left.