Find the Maximum/Minimum Value h=-16t^2+672t

$h = - 16 t^{2} + 672 t$

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 $f (- \frac{b}{2 a})$ .

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

Step 2

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

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

$t = - \frac{672}{2 (- 16)}$

Remove parentheses.

$t = - \frac{672}{2 (- 16)}$

Simplify $- \frac{672}{2 (- 16)}$ .

Cancel the common factor of $672$ and $2$ .

Factor $2$ out of $672$ .

$t = - \frac{2 \cdot 336}{2 \cdot - 16}$

Cancel the common factors.

Factor $2$ out of $2 \cdot - 16$ .

$t = - \frac{2 \cdot 336}{2 (- 16)}$

Cancel the common factor.

$t = - \frac{2 \cdot 336}{2 \cdot - 16}$

Rewrite the expression.

$t = - \frac{336}{- 16}$

Cancel the common factor of $336$ and $- 16$ .

Factor $16$ out of $336$ .

$t = - \frac{16 (21)}{- 16}$

Move the negative one from the denominator of $\frac{21}{- 1}$ .

$t = - (- 1 \cdot 21)$

Multiply $- (- 1 \cdot 21)$ .

Multiply $- 1$ by $21$ .

$t = - - 21$

Multiply $- 1$ by $- 21$ .

$t = 21$

Step 3

Evaluate $f (21)$ .

Replace the variable $t$ with $21$ in the expression.

$f (21) = - 16 {(21)}^{2} + 672 (21)$

Simplify the result.

Simplify each term.

Raise $21$ to the power of $2$ .

$f (21) = - 16 \cdot 441 + 672 (21)$

Multiply $- 16$ by $441$ .

$f (21) = - 7056 + 672 (21)$

Multiply $672$ by $21$ .

$f (21) = - 7056 + 14112$

Add $- 7056$ and $14112$ .

$f (21) = 7056$

The final answer is $7056$ .

$7056$

Step 4

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

$(21, 7056)$

Step 5

Find the Maximum/Minimum Value h=-16t^2+672t

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