Write as a Function of h h=448t-32t^2

$h = 448 t - 32 t^{2}$

Step 1

Rewrite the equation as $448 t - 32 t^{2} = h$ .

$448 t - 32 t^{2} = h$

Step 2

Subtract $h$ from both sides of the equation.

$448 t - 32 t^{2} - h = 0$

Step 3

Use the quadratic formula to find the solutions.

$\frac{- b \pm \sqrt{b^{2} - 4 (a c)}}{2 a}$

Step 4

Substitute the values $a = - 32$ , $b = 448$ , and $c = - h$ into the quadratic formula and solve for $t$ .

$\frac{- 448 \pm \sqrt{448^{2} - 4 \cdot (- 32 \cdot (- h))}}{2 \cdot - 32}$

Step 5

Simplify.

Simplify the numerator.

Raise $448$ to the power of $2$ .

$t = \frac{- 448 \pm \sqrt{200704 - 4 \cdot - 32 \cdot (- 1 h)}}{2 \cdot - 32}$

Multiply $- 4 \cdot - 32 \cdot - 1$ .

Multiply $- 4$ by $- 32$ .

$t = \frac{- 448 \pm \sqrt{200704 + 128 \cdot (- 1 h)}}{2 \cdot - 32}$

Multiply $128$ by $- 1$ .

$t = \frac{- 448 \pm \sqrt{200704 - 128 h}}{2 \cdot - 32}$

Factor $128$ out of $200704 - 128 h$ .

Factor $128$ out of $200704$ .

$t = \frac{- 448 \pm \sqrt{128 (1568) - 128 h}}{2 \cdot - 32}$

Factor $128$ out of $- 128 h$ .

$t = \frac{- 448 \pm \sqrt{128 (1568) + 128 (- h)}}{2 \cdot - 32}$

Factor $128$ out of $128 (1568) + 128 (- h)$ .

$t = \frac{- 448 \pm \sqrt{128 (1568 - h)}}{2 \cdot - 32}$

Rewrite $128 (1568 - h)$ as $8^{2} \cdot (2 (1568 - h))$ .

Factor $64$ out of $128$ .

$t = \frac{- 448 \pm \sqrt{64 (2) (1568 - h)}}{2 \cdot - 32}$

Rewrite $64$ as $8^{2}$ .

$t = \frac{- 448 \pm \sqrt{8^{2} \cdot (2 (1568 - h))}}{2 \cdot - 32}$

Add parentheses.

$t = \frac{- 448 \pm \sqrt{8^{2} \cdot (2 (1568 - h))}}{2 \cdot - 32}$

Pull terms out from under the radical.

$t = \frac{- 448 \pm 8 \sqrt{2 (1568 - h)}}{2 \cdot - 32}$

Multiply $2$ by $- 32$ .

$t = \frac{- 448 \pm 8 \sqrt{2 (1568 - h)}}{- 64}$

Simplify $\frac{- 448 \pm 8 \sqrt{2 (1568 - h)}}{- 64}$ .

$t = \frac{56 \pm \sqrt{2 (1568 - h)}}{8}$

Step 6

Simplify the expression to solve for the $+$ portion of the $\pm$ .

Simplify the numerator.

Raise $448$ to the power of $2$ .

$t = \frac{- 448 \pm \sqrt{200704 - 4 \cdot - 32 \cdot (- 1 h)}}{2 \cdot - 32}$

Multiply $- 4 \cdot - 32 \cdot - 1$ .

Multiply $- 4$ by $- 32$ .

$t = \frac{- 448 \pm \sqrt{200704 + 128 \cdot (- 1 h)}}{2 \cdot - 32}$

Multiply $128$ by $- 1$ .

$t = \frac{- 448 \pm \sqrt{200704 - 128 h}}{2 \cdot - 32}$

Factor $128$ out of $200704 - 128 h$ .

Factor $128$ out of $200704$ .

$t = \frac{- 448 \pm \sqrt{128 (1568) - 128 h}}{2 \cdot - 32}$

Factor $128$ out of $- 128 h$ .

$t = \frac{- 448 \pm \sqrt{128 (1568) + 128 (- h)}}{2 \cdot - 32}$

Factor $128$ out of $128 (1568) + 128 (- h)$ .

$t = \frac{- 448 \pm \sqrt{128 (1568 - h)}}{2 \cdot - 32}$

Rewrite $128 (1568 - h)$ as $8^{2} \cdot (2 (1568 - h))$ .

Factor $64$ out of $128$ .

$t = \frac{- 448 \pm \sqrt{64 (2) (1568 - h)}}{2 \cdot - 32}$

Rewrite $64$ as $8^{2}$ .

$t = \frac{- 448 \pm \sqrt{8^{2} \cdot (2 (1568 - h))}}{2 \cdot - 32}$

Add parentheses.

$t = \frac{- 448 \pm \sqrt{8^{2} \cdot (2 (1568 - h))}}{2 \cdot - 32}$

Pull terms out from under the radical.

$t = \frac{- 448 \pm 8 \sqrt{2 (1568 - h)}}{2 \cdot - 32}$

Multiply $2$ by $- 32$ .

$t = \frac{- 448 \pm 8 \sqrt{2 (1568 - h)}}{- 64}$

Simplify $\frac{- 448 \pm 8 \sqrt{2 (1568 - h)}}{- 64}$ .

$t = \frac{56 \pm \sqrt{2 (1568 - h)}}{8}$

Change the $\pm$ to $+$ .

$t = \frac{56 + \sqrt{2 (1568 - h)}}{8}$

Step 7

Simplify the expression to solve for the $-$ portion of the $\pm$ .

Simplify the numerator.

Raise $448$ to the power of $2$ .

$t = \frac{- 448 \pm \sqrt{200704 - 4 \cdot - 32 \cdot (- 1 h)}}{2 \cdot - 32}$

Multiply $- 4 \cdot - 32 \cdot - 1$ .

Multiply $- 4$ by $- 32$ .

$t = \frac{- 448 \pm \sqrt{200704 + 128 \cdot (- 1 h)}}{2 \cdot - 32}$

Multiply $128$ by $- 1$ .

$t = \frac{- 448 \pm \sqrt{200704 - 128 h}}{2 \cdot - 32}$

Factor $128$ out of $200704 - 128 h$ .

Factor $128$ out of $200704$ .

$t = \frac{- 448 \pm \sqrt{128 (1568) - 128 h}}{2 \cdot - 32}$

Factor $128$ out of $- 128 h$ .

$t = \frac{- 448 \pm \sqrt{128 (1568) + 128 (- h)}}{2 \cdot - 32}$

Factor $128$ out of $128 (1568) + 128 (- h)$ .

$t = \frac{- 448 \pm \sqrt{128 (1568 - h)}}{2 \cdot - 32}$

Rewrite $128 (1568 - h)$ as $8^{2} \cdot (2 (1568 - h))$ .

Factor $64$ out of $128$ .

$t = \frac{- 448 \pm \sqrt{64 (2) (1568 - h)}}{2 \cdot - 32}$

Rewrite $64$ as $8^{2}$ .

$t = \frac{- 448 \pm \sqrt{8^{2} \cdot (2 (1568 - h))}}{2 \cdot - 32}$

Add parentheses.

$t = \frac{- 448 \pm \sqrt{8^{2} \cdot (2 (1568 - h))}}{2 \cdot - 32}$

Pull terms out from under the radical.

$t = \frac{- 448 \pm 8 \sqrt{2 (1568 - h)}}{2 \cdot - 32}$

Multiply $2$ by $- 32$ .

$t = \frac{- 448 \pm 8 \sqrt{2 (1568 - h)}}{- 64}$

Simplify $\frac{- 448 \pm 8 \sqrt{2 (1568 - h)}}{- 64}$ .

$t = \frac{56 \pm \sqrt{2 (1568 - h)}}{8}$

Change the $\pm$ to $-$ .

$t = \frac{56 - \sqrt{2 (1568 - h)}}{8}$

Step 8

The final answer is the combination of both solutions.

$t = \frac{56 + \sqrt{2 (1568 - h)}}{8}$

$t = \frac{56 - \sqrt{2 (1568 - h)}}{8}$

Write as a Function of h h=448t-32t^2

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