Find the Maximum/Minimum Value h(x)=(-32x^2)/((60)^2)+x+210

$h (x) = \frac{- 32 x^{2}}{{(60)}^{2}} + x + 210$

Step 1

Simplify each term.

Raise $60$ to the power of $2$ .

$h (x) = \frac{- 32 x^{2}}{3600} + x + 210$

Cancel the common factor of $- 32$ and $3600$ .

Factor $16$ out of $- 32 x^{2}$ .

$h (x) = \frac{16 (- 2 x^{2})}{3600} + x + 210$

Cancel the common factors.

Factor $16$ out of $3600$ .

$h (x) = \frac{16 (- 2 x^{2})}{16 (225)} + x + 210$

Cancel the common factor.

$h (x) = \frac{16 (- 2 x^{2})}{16 \cdot 225} + x + 210$

Rewrite the expression.

$h (x) = \frac{- 2 x^{2}}{225} + x + 210$

Move the negative in front of the fraction.

$h (x) = - \frac{2 x^{2}}{225} + x + 210$

Step 2

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

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

Step 3

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

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

$x = - \frac{1}{2 (- 0.00\overline{8})}$

Remove parentheses.

$x = - \frac{1}{2 (- 0.00\overline{8})}$

Simplify $- \frac{1}{2 (- 0.00\overline{8})}$ .

Multiply $2$ by $- 0.00\overline{8}$ .

$x = - \frac{1}{- 0.01\overline{7}}$

Divide $1$ by $- 0.01\overline{7}$ .

$x = - - 56.25$

Multiply $- 1$ by $- 56.25$ .

$x = 56.25$

Step 4

Evaluate $f (56.25)$ .

Replace the variable $x$ with $56.25$ in the expression.

$h (56.25) = - \frac{2 {(56.25)}^{2}}{225} + 56.25 + 210$

Simplify the result.

Remove parentheses.

$h (56.25) = - \frac{2 {(56.25)}^{2}}{225} + 56.25 + 210$

Find the common denominator.

Write $56.25$ as a fraction with denominator $1$ .

$h (56.25) = - \frac{2 {(56.25)}^{2}}{225} + \frac{56.25}{1} + 210$

Multiply $\frac{56.25}{1}$ by $\frac{225}{225}$ .

$h (56.25) = - \frac{2 {(56.25)}^{2}}{225} + \frac{56.25}{1} \cdot \frac{225}{225} + 210$

Multiply $\frac{56.25}{1}$ by $\frac{225}{225}$ .

$h (56.25) = - \frac{2 {(56.25)}^{2}}{225} + \frac{56.25 \cdot 225}{225} + 210$

Write $210$ as a fraction with denominator $1$ .

$h (56.25) = - \frac{2 {(56.25)}^{2}}{225} + \frac{56.25 \cdot 225}{225} + \frac{210}{1}$

Multiply $\frac{210}{1}$ by $\frac{225}{225}$ .

$h (56.25) = - \frac{2 {(56.25)}^{2}}{225} + \frac{56.25 \cdot 225}{225} + \frac{210}{1} \cdot \frac{225}{225}$

Multiply $\frac{210}{1}$ by $\frac{225}{225}$ .

$h (56.25) = - \frac{2 {(56.25)}^{2}}{225} + \frac{56.25 \cdot 225}{225} + \frac{210 \cdot 225}{225}$

Combine the numerators over the common denominator.

$h (56.25) = \frac{- 2 {(56.25)}^{2} + 56.25 \cdot 225 + 210 \cdot 225}{225}$

Simplify each term.

Raise $56.25$ to the power of $2$ .

$h (56.25) = \frac{- 2 \cdot 3164.0625 + 56.25 \cdot 225 + 210 \cdot 225}{225}$

Multiply $- 2$ by $3164.0625$ .

$h (56.25) = \frac{- 6328.125 + 56.25 \cdot 225 + 210 \cdot 225}{225}$

Multiply $56.25$ by $225$ .

$h (56.25) = \frac{- 6328.125 + 12656.24999999 + 210 \cdot 225}{225}$

Multiply $210$ by $225$ .

$h (56.25) = \frac{- 6328.125 + 12656.24999999 + 47250}{225}$

Simplify the expression.

Add $- 6328.125$ and $12656.24999999$ .

$h (56.25) = \frac{6328.12499999 + 47250}{225}$

Add $6328.12499999$ and $47250$ .

$h (56.25) = \frac{53578.12499999}{225}$

Divide $53578.12499999$ by $225$ .

$h (56.25) = 238.125$

The final answer is $238.125$ .

$238.125$

Step 5

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

$(56.25, 238.125)$

Step 6

Find the Maximum/Minimum Value h(x)=(-32x^2)/((60)^2)+x+210

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