Respuesta :
The two numbers are 120 and 60
Let's call the two numbers x and y. The first sentence translates into
[tex] x+2y = 240 [/tex]
from which we can derive
[tex] x = 240-2y [/tex]
So, their product can be written as
[tex] xy = (240-2y)y = 240y-2y^2 [/tex]
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 [tex] ax^2+bx+c [/tex], its extreme point is located at [tex] x = \frac{-b}{2a} [/tex].
So, in your case, since [tex] a = -2 [/tex] and [tex] b = 240 [/tex], the maximum is located at
[tex] y = \frac{-240}{-4} = 60 [/tex]
Now that we know y, we can deduce the value of x:
[tex] x = 240-2y \implies x = 240-120 = 120[/tex]
So, the two numbers are 120 and 60
We will see that the two numbers that meet the condition and maximize the product are x = 120 and y = 60.
How to find the two numbers?
Let's say that our two numbers are x and y.
We want that the sum of the first number and twice the second is 240, then:
x + 2y = 240
We also want to maximize the product: P = x*y
To do it, first we isolate one of the variables in the first equation, I will isolate x:
x = 240 - 2y
Now we replace this on the product equation:
P = (240 - 2y)*y
P = 240y - 2y^2
Notice that this is a quadratic equation of negative leading coefficient, this means that the maximum is on the vertex.
Remember that for a equation like:
y = a*x^2 + b*x + c
The x-value of the vertex is:
x = -b/2a
So in our case, the vertex will be at:
y = -(240)/(2*-2) = 240/4 = 60
This is the value of y that maximizes the product, to get the value of x we use:
x = 240 - 2*y = 240 - 2*60 = 120
Then the two numbers are:
x = 120 and y = 60.
If you want to learn more about maximization, you can read:
https://brainly.com/question/19819849
