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The positive numbers
x and
a-x have a sum of
a. What is
x in terms of
a if the product
x*(a-x) is a maximum?
Choose 1 answer:
(A)
(a)/(2)
(B)
sqrt((a)/(2))
(C)
sqrta
(D)
a
E There is no
x that would produce a maximum product

The positive numbers $x$ and $a-x$ have a sum of $a$ . What is $x$ in terms of $a$ if the product $x \cdot(a-x)$ is a maximum? $\newline$ Choose $1$ answer: $\newline$ (A) $\frac{a}{2}$ $\newline$ (B) $\sqrt{\frac{a}{2}}$ $\newline$ (C) $\sqrt{a}$ $\newline$ (D) $a$ $\newline$ (E) There is no $x$ that would produce a maximum product

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Simplify radical expressions with variables II

Full solution

Q. The positive numbers

x

and

a-x

have a sum of

a

. What is

x

in terms of

a

if the product

x \cdot(a-x)

is a maximum?

\newline

Choose

1

answer:

\newline

(A)

\frac{a}{2}

\newline

(B)

\sqrt{\frac{a}{2}}

\newline

(C)

\sqrt{a}

\newline

(D)

a

\newline

(E) There is no

x

that would produce a maximum product

Equation Simplification: We know that $x + (a - x) = a$ . This simplifies to $x + a - x = a$ , which means $x$ cancels out and we're left with $a = a$ , which is true but doesn't help us find $x$ .
Maximizing Product: To maximize the product $x(a-x)$ , we can use the fact that for a fixed sum, the product of two numbers is maximized when the numbers are equal. So, we set $x = a - x$ .
Solving for x: Solving for x gives us $x = a - x$ , which simplifies to $2x = a$ .
Isolating $x$ : Divide both sides by $2$ to isolate $x$ , we get $x = \frac{a}{2}$ .

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\newline

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\newline

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