Question

What must (a) be if the perimeter of the figure should be as small as possible?

Should I just guess some numbers and see which gives the smallest perimeter? Or are there any other ways to solve this problem?

Answer 1

`a =2`

Explanation:

The first thing to notice is the height and width of your rectangle.

We know that the first vertex is drawn `a` units from the origin, so the width will be `a`. Then you can see that the height is `f(a)`, or the value of the function at `x = a`. But since the function we are considering is `f(x) = 4/x`, we can see that `f(a) = 4/a`. The expression for perimeter will therefore be

`2(4/a) + 2a = P`

`8/a + 2a = P`

Now differentiate with respect to `P`.

`P' = -8/a^2 + 2`

This will have critical numbers when the derivative equals `0`.

`0 = -8/a^2 + 2`

`-2 = -8/a^2`

`a^2 = 4`

`a = +-2`

The derivative is negative from `(-2, 2)` and positive from `(2, oo)` and `(-oo, 2)`.

Since the transition from negative to positive occurs at `x = 2`, there is a minimum perimeter at `x =2`. This minimum perimeter has value `8/2 + 2(2) = 8` units.

Hopefully this helps!