Find the Maximum/Minimum Value R(p)=-2.5p^2+450p

$R (p) = - 2.5 p^{2} + 450 p$

Step 1

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

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

Step 2

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

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

$p = - \frac{450}{2 (- 2.5)}$

Remove parentheses.

$p = - \frac{450}{2 (- 2.5)}$

Simplify $- \frac{450}{2 (- 2.5)}$ .

Cancel the common factor of $450$ and $2$ .

Factor $2$ out of $450$ .

$p = - \frac{2 \cdot 225}{2 \cdot - 2.5}$

Cancel the common factors.

Factor $2$ out of $2 \cdot - 2.5$ .

$p = - \frac{2 \cdot 225}{2 (- 2.5)}$

Cancel the common factor.

$p = - \frac{2 \cdot 225}{2 \cdot - 2.5}$

Rewrite the expression.

$p = - \frac{225}{- 2.5}$

Simplify the expression.

Divide $225$ by $- 2.5$ .

$p = - - 90$

Multiply $- 1$ by $- 90$ .

$p = 90$

Step 3

Evaluate $f (90)$ .

Replace the variable $p$ with $90$ in the expression.

$R (90) = - 2.5 {(90)}^{2} + 450 (90)$

Simplify the result.

Simplify each term.

Raise $90$ to the power of $2$ .

$R (90) = - 2.5 \cdot 8100 + 450 (90)$

Multiply $- 2.5$ by $8100$ .

$R (90) = - 20250 + 450 (90)$

Multiply $450$ by $90$ .

$R (90) = - 20250 + 40500$

Add $- 20250$ and $40500$ .

$R (90) = 20250$

The final answer is $20250$ .

$20250$

Step 4

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

$(90, 20250)$

Step 5

Find the Maximum/Minimum Value R(p)=-2.5p^2+450p

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