Let's solve this problem step-by-step.
1. Define the Problem:
- Let's denote one of the numbers as [tex]\( x \)[/tex].
- Since the sum of the two numbers is 48, the other number is [tex]\( 48 - x \)[/tex].
2. Write the Objective Function:
- The product [tex]\( P \)[/tex] of these two numbers is given by:
[tex]\[
P = x \cdot (48 - x)
\][/tex]
Which simplifies to:
[tex]\[
P = 48x - x^2
\][/tex]
3. Determine the Interval:
- Since we are dealing with nonnegative real numbers, [tex]\( x \)[/tex] must satisfy:
[tex]\[
0 \leq x \leq 48
\][/tex]
Therefore, the interval of interest for [tex]\( x \)[/tex] is [tex]\( [0, 48] \)[/tex].
4. Find the Maximum Product:
- The function [tex]\( P = 48x - x^2 \)[/tex] is a quadratic function in the form [tex]\( P = -x^2 + 48x \)[/tex].
- The maximum value of a quadratic function [tex]\( ax^2 + bx + c \)[/tex] occurs at the vertex, which is given by [tex]\( x = -\frac{b}{2a} \)[/tex].
- Here, [tex]\( a = -1 \)[/tex] and [tex]\( b = 48 \)[/tex], so the vertex is at:
[tex]\[
x = -\frac{48}{2 \cdot -1} = 24
\][/tex]
5. Conclusion:
- The value of [tex]\( x \)[/tex] that maximizes the product is 24.
- The corresponding other number is [tex]\( 48 - 24 = 24 \)[/tex].
Therefore, the two nonnegative real numbers that sum to 48 and have the largest possible product are:
[tex]\[
24, 24
\][/tex]
In summary:
- The objective function in terms of [tex]\( x \)[/tex] is [tex]\( P = 48x - x^2 \)[/tex].
- The interval of interest is [tex]\([0, 48]\)[/tex].
- The numbers that have the largest possible product are [tex]\( 24 \)[/tex] and [tex]\( 24 \)[/tex].
