Find the Maximum/Minimum Value h=-16t^2+176t+1632

$h = - 16 t^{2} + 176 t + 1632$

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 $b$ .

$t = - \frac{176}{2 (- 16)}$

Remove parentheses.

$t = - \frac{176}{2 (- 16)}$

Simplify $- \frac{176}{2 (- 16)}$ .

Cancel the common factor of $176$ and $2$ .

Factor $2$ out of $176$ .

$t = - \frac{2 \cdot 88}{2 \cdot - 16}$

Cancel the common factors.

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

$t = - \frac{2 \cdot 88}{2 (- 16)}$

Cancel the common factor.

$t = - \frac{2 \cdot 88}{2 \cdot - 16}$

Rewrite the expression.

$t = - \frac{88}{- 16}$

Cancel the common factor of $88$ and $- 16$ .

Factor $8$ out of $88$ .

$t = - \frac{8 (11)}{- 16}$

Cancel the common factors.

Factor $8$ out of $- 16$ .

$t = - \frac{8 \cdot 11}{8 \cdot - 2}$

Cancel the common factor.

$t = - \frac{8 \cdot 11}{8 \cdot - 2}$

Rewrite the expression.

$t = - \frac{11}{- 2}$

Move the negative in front of the fraction.

$t = - - \frac{11}{2}$

Multiply $- - \frac{11}{2}$ .

Multiply $- 1$ by $- 1$ .

$t = 1 (\frac{11}{2})$

Multiply $\frac{11}{2}$ by $1$ .

$t = \frac{11}{2}$

Step 3

Evaluate $f (\frac{11}{2})$ .

Replace the variable $t$ with $\frac{11}{2}$ in the expression.

$f (\frac{11}{2}) = - 16 {(\frac{11}{2})}^{2} + 176 (\frac{11}{2}) + 1632$

Simplify the result.

Simplify each term.

Apply the product rule to $\frac{11}{2}$ .

$f (\frac{11}{2}) = - 16 \frac{11^{2}}{2^{2}} + 176 (\frac{11}{2}) + 1632$

Raise $11$ to the power of $2$ .

$f (\frac{11}{2}) = - 16 \frac{121}{2^{2}} + 176 (\frac{11}{2}) + 1632$

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

$f (\frac{11}{2}) = - 16 (\frac{121}{4}) + 176 (\frac{11}{2}) + 1632$

Cancel the common factor of $4$ .

Factor $4$ out of $- 16$ .

$f (\frac{11}{2}) = 4 (- 4) (\frac{121}{4}) + 176 (\frac{11}{2}) + 1632$

Cancel the common factor.

$f (\frac{11}{2}) = 4 \cdot (- 4 (\frac{121}{4})) + 176 (\frac{11}{2}) + 1632$

Rewrite the expression.

$f (\frac{11}{2}) = - 4 \cdot 121 + 176 (\frac{11}{2}) + 1632$

Multiply $- 4$ by $121$ .

$f (\frac{11}{2}) = - 484 + 176 (\frac{11}{2}) + 1632$

Cancel the common factor of $2$ .

Factor $2$ out of $176$ .

$f (\frac{11}{2}) = - 484 + 2 (88) (\frac{11}{2}) + 1632$

Cancel the common factor.

$f (\frac{11}{2}) = - 484 + 2 \cdot (88 (\frac{11}{2})) + 1632$

Rewrite the expression.

$f (\frac{11}{2}) = - 484 + 88 \cdot 11 + 1632$

Multiply $88$ by $11$ .

$f (\frac{11}{2}) = - 484 + 968 + 1632$

Simplify by adding numbers.

Add $- 484$ and $968$ .

$f (\frac{11}{2}) = 484 + 1632$

Add $484$ and $1632$ .

$f (\frac{11}{2}) = 2116$

The final answer is $2116$ .

$2116$

Step 4

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

$(\frac{11}{2}, 2116)$

Step 5

Find the Maximum/Minimum Value h=-16t^2+176t+1632

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