Question

What are two positive numbers whose sum of the first number squared and the second number is 54 and the product is a maximum?

Answer 1

`3sqrt(2) and 36`

Explanation:

Let the numbers be `w` and `x`.

`x^2 + w = 54`

We want to find

`P = wx`

We can rearrange the original equation to be `w = 54 - x^2`. Substituting we get

`P = (54 - x^2)x`

`P = 54x - x^3`

Now take the derivative with respect to `x`.

`P' = 54 - 3x^2`

Let `P' = 0`.

`0 = 54 - 3x^2`

`3x^2 =54`

`x =+-sqrt(18) =+- 3sqrt(2)`

But since we're given that the numbers have to be positive, we can only accept `x =3sqrt(2)`. Now we verify that this is indeed a maximum.

At `x = 3`, the derivative is positive.

At `x = 5`, the derivative is negative.

Therefore, `x =3sqrt(2)` and `54 -(3sqrt(2))^2 = 36` give a maximum product when multiplied.

Hopefully this helps!