Question

find two positive real numbers whose prodcut is a maximum. 1) The sum is 10. 2) Sum of first and twice the second is 24. Please explain not just tell answer. Thanks

Answer 1

For integers, you only need to tabulate and find the maximum (5*5>4*6>3*7...)
For real numbers, and to solve it correctly, you need to use calculus, or the properties of parabolas.
Which of these are you working on?

Answer 2

I can only assume where working on properties of parabolas, because where dealing with parabola graphs now but i missed a whole week of class and just collected the homework.

Answer 3

If you're working on parabolas, you can try to formulate the problem in the form of a quadratic equation, and use completing squares to find the vertex, which is where the maximum or minimum is.

Answer 4

Could you tell me how to go about doing so to solve it anyway..... I have the answers for both i just don't quite understand The sum is 10 one.. The answer is 55,55.

Answer 5

I will solve the first one. You can try the second one in a similar way.

The canonical form of a quadratic equation is
y=a(x-h)^2+k
where a is a constant,
(h,k) are the coordinates of the vertex

Let x be one of the numbers, then 10-x is the other (sum=10).
The product of the two numbers is therefore:
y=x(10-x)
=10x-x^2
Completing squares
=-1(x^2-10x+25) + 25
=-(x-5)^2+25
so the vertex is at (5,25),
which means that the numbers are x=5, and x=10-5=5 (both 5).

For the second problem, the sum of one number and twice the other is 24, so one number is x, and the other is 24-2x.

Answer 6

In my textbook it says the answers 55,55 as the two numbers that's the reason I am confused/ I'm a lost cause for this though right now though I appreciate you trying to help me so I award you. :]