Find the Maximum/Minimum Value r(p)=-10p^2+130p

$r (p) = - 10 p^{2} + 130 p$

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 $b$ .

$p = - \frac{130}{2 (- 10)}$

Remove parentheses.

$p = - \frac{130}{2 (- 10)}$

Simplify $- \frac{130}{2 (- 10)}$ .

Cancel the common factor of $130$ and $2$ .

Factor $2$ out of $130$ .

$p = - \frac{2 \cdot 65}{2 \cdot - 10}$

Cancel the common factors.

Factor $2$ out of $2 \cdot - 10$ .

$p = - \frac{2 \cdot 65}{2 (- 10)}$

Cancel the common factor.

$p = - \frac{2 \cdot 65}{2 \cdot - 10}$

Rewrite the expression.

$p = - \frac{65}{- 10}$

Cancel the common factor of $65$ and $- 10$ .

Factor $5$ out of $65$ .

$p = - \frac{5 (13)}{- 10}$

Cancel the common factors.

Factor $5$ out of $- 10$ .

$p = - \frac{5 \cdot 13}{5 \cdot - 2}$

Cancel the common factor.

$p = - \frac{5 \cdot 13}{5 \cdot - 2}$

Rewrite the expression.

$p = - \frac{13}{- 2}$

Move the negative in front of the fraction.

$p = - - \frac{13}{2}$

Multiply $- - \frac{13}{2}$ .

Multiply $- 1$ by $- 1$ .

$p = 1 (\frac{13}{2})$

Multiply $\frac{13}{2}$ by $1$ .

$p = \frac{13}{2}$

Step 3

Evaluate $f (\frac{13}{2})$ .

Replace the variable $p$ with $\frac{13}{2}$ in the expression.

$r (\frac{13}{2}) = - 10 {(\frac{13}{2})}^{2} + 130 (\frac{13}{2})$

Simplify the result.

Simplify each term.

Apply the product rule to $\frac{13}{2}$ .

$r (\frac{13}{2}) = - 10 \frac{13^{2}}{2^{2}} + 130 (\frac{13}{2})$

Raise $13$ to the power of $2$ .

$r (\frac{13}{2}) = - 10 \frac{169}{2^{2}} + 130 (\frac{13}{2})$

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

$r (\frac{13}{2}) = - 10 (\frac{169}{4}) + 130 (\frac{13}{2})$

Cancel the common factor of $2$ .

Factor $2$ out of $- 10$ .

$r (\frac{13}{2}) = 2 (- 5) (\frac{169}{4}) + 130 (\frac{13}{2})$

Factor $2$ out of $4$ .

$r (\frac{13}{2}) = 2 \cdot (- 5 \frac{169}{2 \cdot 2}) + 130 (\frac{13}{2})$

Cancel the common factor.

$r (\frac{13}{2}) = 2 \cdot (- 5 \frac{169}{2 \cdot 2}) + 130 (\frac{13}{2})$

Rewrite the expression.

$r (\frac{13}{2}) = - 5 (\frac{169}{2}) + 130 (\frac{13}{2})$

Combine $- 5$ and $\frac{169}{2}$ .

$r (\frac{13}{2}) = \frac{- 5 \cdot 169}{2} + 130 (\frac{13}{2})$

Multiply $- 5$ by $169$ .

$r (\frac{13}{2}) = \frac{- 845}{2} + 130 (\frac{13}{2})$

Move the negative in front of the fraction.

$r (\frac{13}{2}) = - \frac{845}{2} + 130 (\frac{13}{2})$

Cancel the common factor of $2$ .

Factor $2$ out of $130$ .

$r (\frac{13}{2}) = - \frac{845}{2} + 2 (65) (\frac{13}{2})$

Cancel the common factor.

$r (\frac{13}{2}) = - \frac{845}{2} + 2 \cdot (65 (\frac{13}{2}))$

Rewrite the expression.

$r (\frac{13}{2}) = - \frac{845}{2} + 65 \cdot 13$

Multiply $65$ by $13$ .

$r (\frac{13}{2}) = - \frac{845}{2} + 845$

To write $845$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$r (\frac{13}{2}) = - \frac{845}{2} + 845 \cdot \frac{2}{2}$

Combine $845$ and $\frac{2}{2}$ .

$r (\frac{13}{2}) = - \frac{845}{2} + \frac{845 \cdot 2}{2}$

Combine the numerators over the common denominator.

$r (\frac{13}{2}) = \frac{- 845 + 845 \cdot 2}{2}$

Simplify the numerator.

Multiply $845$ by $2$ .

$r (\frac{13}{2}) = \frac{- 845 + 1690}{2}$

Add $- 845$ and $1690$ .

$r (\frac{13}{2}) = \frac{845}{2}$

The final answer is $\frac{845}{2}$ .

$\frac{845}{2}$

Step 4

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

$(\frac{13}{2}, \frac{845}{2})$

Step 5

Find the Maximum/Minimum Value r(p)=-10p^2+130p

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