Find two positive numbers that satisfy the given requirements. The sum of the first number squared and the second number is 60 and the product is a maximum?

Question

Find two positive numbers that satisfy the given requirements. The sum of the first number squared and the second number is 60 and the product is a maximum?

1 Answer

Impact of this question

20431 views around the world

You can reuse this answer
Creative Commons License

Answer 1 · 2017-04-12T03:50:31

The numbers are $40$ and $2sqrt(5)$ . I know these aren't integers (and $sqrt(5)$ isn't a rational number), but this is the most logical solution to this problem.

Explanation:

Let the numbers be $x$ and $y$ .

$x^2 + y = 60 -> y = 60 - x^2$

The product will be $P = xy$ . Substituting from the first equation, we get:

$P = (60 - x^2)x$

$P = -x^3 + 60x$

We now find the derivative with respect to $x$ .

$P' = -3x^2 + 60$

Now determine the critical numbers, which will occur when $P' = 0$ .

$0 = -3x^2 + 60$

$0 = -3(x^2 - 20)$

$x = +- sqrt(20)$

$x= +- 2sqrt(5)$

We must check to make sure $x = + 2sqrt(5)$ is indeed a maximum.

Test point $1$ : $x = 4$

$P'(4) = -3(4)^2 + 60 = "positive"$

Test point $2$ : $x = 5$

$P'(5) = -3(5)^2 + 60 = "negative"$

By increasing/decreasing rules, we can conclude that $2sqrt(5)$ is a local maximum (this function has no absolute maximum).

This means that $y = 60 - (2sqrt(5))^2 = 40$ .

Hopefully this helps!

Science

Math

Humanities

... and beyond

Find two positive numbers that satisfy the given requirements. The sum of the first number squared and the second number is 60 and the product is a maximum?

1 Answer

Explanation:

Related questions

Impact of this question