Contents

Exercise 1

A square and a triangle have the same area. Which shape has the greatest perimeter?

Hint
It may be useful to remember the following formula:

Arithmetic Mean-Geometric Mean Inequality
Let $$x_{1},x_{2}, \dots x_{n}$$ be non-negative real numbers; then:

$\dfrac {x_{1}+ x_{2} + \cdots x_{n}}{n} \ge \sqrt[n]{ x_{1}x_{2} \cdots x_{n}}$

Solution
The triangle has the greatest perimeter.

Exercise 2

Let $$S$$ be a sphere of fixed radius $$R$$. Determine the height of the inscribed cone of maximum volume.

Solve also without using calculus (first derivative test).

Solution: [ $$h=\frac{4R}{3}$$ ]

Exercise 3

Let $$T$$ be the set of all the triangles $$\triangle ABC$$, which have a fixed angle $$\alpha$$ in the vertex $$A$$ and a fixed area $$S$$. Show that the one with the shortest base $$BC$$ is an isosceles triangle.

Solve also without using calculus (first derivative test).
It may be useful to remember the following formulas:

Carnot’s theorem (cosine formula)
Let $$\triangle ABC$$ be a triangle with sides of length $$a,b,c$$; then

$a^2 = b^2 + c^2 – 2bc \cos \alpha$

where $$\alpha$$ is opposite to the side of length $$a$$. Similar formulas apply to the other two sides $$b,c$$.

Area of ​​a triangle
The area of ​​a triangle with side lengths $$a,b,c$$ is:

$S = \frac{bc \sin \alpha}{2}$

where $$\alpha$$ is opposite to the side of length $$a$$. Similar formulas apply to the other two angles.