Application prob: find numbers w/ sum of 60, product is max

[incarnate]

I'm having problems formulating the function that relates to the vertex of the parabola (were it graphed).

Problem: Of all the pairs of numbers whose sum is 60, which pair has a maximum product?

 

[Deleted member 4993]

Re: Application problem

I'm having problems formulating the function that relates to the vertex of the parabola (were it graphed).

Problem: Of all the pairs of numbers whose sum is 60, which pair has a maximum product?

Please show us your work - indicating exactly where you are stuck, so that we know where to begin to help you.

 

[stapel]

Of all the pairs of numbers whose sum is 60, which pair has a maximum product?

What have you tried? How far did you get? Where are you stuck? You picked a variable for one of the numbers, used this (along with the value of their sum) to create an expression for the other number, multiplied the variable and the expression, found the vertex, and... then what?

Please be complete. Thank you!

Eliz.

 

[soroban]

Hello, incarnate!

Okay, I'll get you started . . .

Of all the pairs of numbers whose sum is 60, which pair has a maximum product?

Let \(\displaystyle x\) = one number.
Then \(\displaystyle 60 - x\) is the other number.

Their product is: .\(\displaystyle P \;=\;x(60-x)\)

Can you determine the maximum value of \(\displaystyle P\) ?

 

[fasteddie65]

The maximum (or minimum) value of a quadratic expression occurs at the vertex of the graph of the function.
Do you remember that the x-coordinate of the vertex is x = -b/(2a).
In this case, x = (-60)/(2 • -1) = -60/-2 = 30.
Since a < 0, the value is a maximum.
P = x(60 - x) = 30(60 - 30) = 30 • 30 = 900
The two numbers are both 30.