The sum of two numbers is 24. Find the maximum value of their product.

Question

asked Nov 13, 2023 74.7k views

1 Answer

Al Grant · Answer 1 · 2023-11-19T13:12:15+0000

Let x and y be the two numbers we are looking for. We are told that the sum is 24. So we have the equation

we want to calculate the maximum value of the product of both numbers, that is

$x\cdot y$

From the first equation we could replace the value of y, so we have that

$x(24\text{ -x\rparen=24x- x}^2$

which is a parabolla. Recall that the general form of a parabolla is given by the equation

$y=a(x\text{ -h\rparen}^2+k$

where (h,k) is the vertex of the parabolla. In this case, k would be the maximum or minimum value of the parabolla.

We start by factoring out the -1. So we get

$y=\text{ - \lparen x}^2\text{ -24x\rparen}$

now, we will complete the square inside the parentheses. Note that if

$(a\text{ -b\rparen}^2=a^2\text{ - 2ab + b}^2$

and we let a = x we have that

$(x\text{ -b\rparen}^2=x^2\text{ -2bx+b}^2$

If we compare this to the expression x^2 -24x, we can see that -2b=-24. So we have

$\begin{gathered} \text{ -2b= -24} \\ b=\frac{\text{ -24}}{\placeholder{⬚}\text{ -2}} \\ b=12 \end{gathered}$

So, we will add and subtract 12² so we get

$y=\text{ -\lparen x}^2\text{ -24x+12}^2\text{ -12}^2)=\text{ -\lparen x}^2\text{ -24x+12}^2)\text{ +12}^2$

which is equivalent to

$y=\text{ -\lparen x -12\rparen}^2+12^2$

by comparison, we can see that in here the value of k is 12^2. THat is

which is the maximum value of the product. So the correction option is the third option.