Question

Find two positive numbers satisfying the given requirements. the sum of the first and twice the second is 240 and the product is a maximum.

Answer 1

The two numbers are 120 and 60

Let's call the two numbers x and y. The first sentence translates into

$x+2y = 240$

from which we can derive

$x = 240-2y$

So, their product can be written as

$xy = (240-2y)y = 240y-2y^2$

This expression is a quadratic polynomial with negative leading coefficient, so it represents a parabola concave down. So, the vertex of the parabola is its maximum, which we can find as usual: given the parabola $ax^2+bx+c$, its extreme point is located at $x = \frac{-b}{2a}$.

So, in your case, since $a = -2$ and $b = 240$, the maximum is located at

$y = \frac{-240}{-4} = 60$

Now that we know y, we can deduce the value of x:

$x = 240-2y \implies x = 240-120 = 120$

So, the two numbers are 120 and 60