Find the Maximum/Minimum Value A(w)=-w(w-80)

$A (w) = - w (w - 80)$

Step 1

Apply the distributive property.

$A (w) = - w \cdot w - w \cdot - 80$

Step 2

Multiply $w$ by $w$ by adding the exponents.

Move $w$ .

$A (w) = - (w \cdot w) - w \cdot - 80$

Multiply $w$ by $w$ .

$A (w) = - w^{2} - w \cdot - 80$

Step 3

Multiply $- 80$ by $- 1$ .

$A (w) = - w^{2} + 80 w$

Step 4

The maximum of a quadratic function occurs at $x = - \frac{b}{2 a}$ . If $a$ is negative, the maximum value of the function is $f (- \frac{b}{2 a})$ .

$f_{max}$ $w = a w^{2} + b w + c$ occurs at $w = - \frac{b}{2 a}$

Step 5

Find the value of $w = - \frac{b}{2 a}$ .

Substitute in the values of $a$ and $b$ .

$w = - \frac{80}{2 (- 1)}$

Remove parentheses.

$w = - \frac{80}{2 (- 1)}$

Simplify $- \frac{80}{2 (- 1)}$ .

Cancel the common factor of $80$ and $2$ .

Factor $2$ out of $80$ .

$w = - \frac{2 \cdot 40}{2 \cdot - 1}$

Move the negative one from the denominator of $\frac{40}{- 1}$ .

$w = - (- 1 \cdot 40)$

Multiply $- (- 1 \cdot 40)$ .

Multiply $- 1$ by $40$ .

$w = - - 40$

Multiply $- 1$ by $- 40$ .

$w = 40$

Step 6

Evaluate $f (40)$ .

Replace the variable $w$ with $40$ in the expression.

$A (40) = - {(40)}^{2} + 80 (40)$

Simplify the result.

Simplify each term.

Raise $40$ to the power of $2$ .

$A (40) = - 1 \cdot 1600 + 80 (40)$

Multiply $- 1$ by $1600$ .

$A (40) = - 1600 + 80 (40)$

Multiply $80$ by $40$ .

$A (40) = - 1600 + 3200$

Add $- 1600$ and $3200$ .

$A (40) = 1600$

The final answer is $1600$ .

$1600$

Step 7

Use the $x$ and $y$ values to find where the maximum occurs.

$(40, 1600)$

Step 8

Find the Maximum/Minimum Value A(w)=-w(w-80)

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