Find the Maximum/Minimum Value h=-16t^2+224t+1152

$h = - 16 t^{2} + 224 t + 1152$

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 $b$ .

$t = - \frac{224}{2 (- 16)}$

Remove parentheses.

$t = - \frac{224}{2 (- 16)}$

Simplify $- \frac{224}{2 (- 16)}$ .

Cancel the common factor of $224$ and $2$ .

Factor $2$ out of $224$ .

$t = - \frac{2 \cdot 112}{2 \cdot - 16}$

Cancel the common factors.

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

$t = - \frac{2 \cdot 112}{2 (- 16)}$

Cancel the common factor.

$t = - \frac{2 \cdot 112}{2 \cdot - 16}$

Rewrite the expression.

$t = - \frac{112}{- 16}$

Cancel the common factor of $112$ and $- 16$ .

Factor $16$ out of $112$ .

$t = - \frac{16 (7)}{- 16}$

Move the negative one from the denominator of $\frac{7}{- 1}$ .

$t = - (- 1 \cdot 7)$

Multiply $- (- 1 \cdot 7)$ .

Multiply $- 1$ by $7$ .

$t = - - 7$

Multiply $- 1$ by $- 7$ .

$t = 7$

Step 3

Evaluate $f (7)$ .

Replace the variable $t$ with $7$ in the expression.

$f (7) = - 16 {(7)}^{2} + 224 (7) + 1152$

Simplify the result.

Simplify each term.

Raise $7$ to the power of $2$ .

$f (7) = - 16 \cdot 49 + 224 (7) + 1152$

Multiply $- 16$ by $49$ .

$f (7) = - 784 + 224 (7) + 1152$

Multiply $224$ by $7$ .

$f (7) = - 784 + 1568 + 1152$

Simplify by adding numbers.

Add $- 784$ and $1568$ .

$f (7) = 784 + 1152$

Add $784$ and $1152$ .

$f (7) = 1936$

The final answer is $1936$ .

$1936$

Step 4

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

$(7, 1936)$

Step 5

Find the Maximum/Minimum Value h=-16t^2+224t+1152

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