Write as a Function of h h=-16t^2+144t+160

$h = - 16 t^{2} + 144 t + 160$

Step 1

Rewrite the equation as $- 16 t^{2} + 144 t + 160 = h$ .

$- 16 t^{2} + 144 t + 160 = h$

Step 2

Subtract $h$ from both sides of the equation.

$- 16 t^{2} + 144 t + 160 - h = 0$

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 $a = - 16$ , $b = 144$ , and $c = 160 - h$ into the quadratic formula and solve for $t$ .

$\frac{- 144 \pm \sqrt{144^{2} - 4 \cdot (- 16 \cdot (160 - h))}}{2 \cdot - 16}$

Step 5

Simplify.

Simplify the numerator.

Raise $144$ to the power of $2$ .

$t = \frac{- 144 \pm \sqrt{20736 - 4 \cdot - 16 \cdot (160 - h)}}{2 \cdot - 16}$

Multiply $- 4$ by $- 16$ .

$t = \frac{- 144 \pm \sqrt{20736 + 64 \cdot (160 - h)}}{2 \cdot - 16}$

Apply the distributive property.

$t = \frac{- 144 \pm \sqrt{20736 + 64 \cdot 160 + 64 (- h)}}{2 \cdot - 16}$

Multiply $64$ by $160$ .

$t = \frac{- 144 \pm \sqrt{20736 + 10240 + 64 (- h)}}{2 \cdot - 16}$

Multiply $- 1$ by $64$ .

$t = \frac{- 144 \pm \sqrt{20736 + 10240 - 64 h}}{2 \cdot - 16}$

Add $20736$ and $10240$ .

$t = \frac{- 144 \pm \sqrt{30976 - 64 h}}{2 \cdot - 16}$

Factor $64$ out of $30976 - 64 h$ .

Factor $64$ out of $30976$ .

$t = \frac{- 144 \pm \sqrt{64 (484) - 64 h}}{2 \cdot - 16}$

Factor $64$ out of $- 64 h$ .

$t = \frac{- 144 \pm \sqrt{64 (484) + 64 (- h)}}{2 \cdot - 16}$

Factor $64$ out of $64 (484) + 64 (- h)$ .

$t = \frac{- 144 \pm \sqrt{64 (484 - h)}}{2 \cdot - 16}$

Rewrite $64 (484 - h)$ as $8^{2} (22^{2} - h)$ .

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

$t = \frac{- 144 \pm \sqrt{8^{2} (484 - h)}}{2 \cdot - 16}$

Rewrite $484$ as $22^{2}$ .

$t = \frac{- 144 \pm \sqrt{8^{2} (22^{2} - h)}}{2 \cdot - 16}$

Pull terms out from under the radical.

$t = \frac{- 144 \pm 8 \sqrt{22^{2} - h}}{2 \cdot - 16}$

Raise $22$ to the power of $2$ .

$t = \frac{- 144 \pm 8 \sqrt{484 - h}}{2 \cdot - 16}$

Multiply $2$ by $- 16$ .

$t = \frac{- 144 \pm 8 \sqrt{484 - h}}{- 32}$

Simplify $\frac{- 144 \pm 8 \sqrt{484 - h}}{- 32}$ .

$t = \frac{18 \pm \sqrt{484 - h}}{4}$

Step 6

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

Simplify the numerator.

Raise $144$ to the power of $2$ .

$t = \frac{- 144 \pm \sqrt{20736 - 4 \cdot - 16 \cdot (160 - h)}}{2 \cdot - 16}$

Multiply $- 4$ by $- 16$ .

$t = \frac{- 144 \pm \sqrt{20736 + 64 \cdot (160 - h)}}{2 \cdot - 16}$

Apply the distributive property.

$t = \frac{- 144 \pm \sqrt{20736 + 64 \cdot 160 + 64 (- h)}}{2 \cdot - 16}$

Multiply $64$ by $160$ .

$t = \frac{- 144 \pm \sqrt{20736 + 10240 + 64 (- h)}}{2 \cdot - 16}$

Multiply $- 1$ by $64$ .

$t = \frac{- 144 \pm \sqrt{20736 + 10240 - 64 h}}{2 \cdot - 16}$

Add $20736$ and $10240$ .

$t = \frac{- 144 \pm \sqrt{30976 - 64 h}}{2 \cdot - 16}$

Factor $64$ out of $30976 - 64 h$ .

Factor $64$ out of $30976$ .

$t = \frac{- 144 \pm \sqrt{64 (484) - 64 h}}{2 \cdot - 16}$

Factor $64$ out of $- 64 h$ .

$t = \frac{- 144 \pm \sqrt{64 (484) + 64 (- h)}}{2 \cdot - 16}$

Factor $64$ out of $64 (484) + 64 (- h)$ .

$t = \frac{- 144 \pm \sqrt{64 (484 - h)}}{2 \cdot - 16}$

Rewrite $64 (484 - h)$ as $8^{2} (22^{2} - h)$ .

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

$t = \frac{- 144 \pm \sqrt{8^{2} (484 - h)}}{2 \cdot - 16}$

Rewrite $484$ as $22^{2}$ .

$t = \frac{- 144 \pm \sqrt{8^{2} (22^{2} - h)}}{2 \cdot - 16}$

Pull terms out from under the radical.

$t = \frac{- 144 \pm 8 \sqrt{22^{2} - h}}{2 \cdot - 16}$

Raise $22$ to the power of $2$ .

$t = \frac{- 144 \pm 8 \sqrt{484 - h}}{2 \cdot - 16}$

Multiply $2$ by $- 16$ .

$t = \frac{- 144 \pm 8 \sqrt{484 - h}}{- 32}$

Simplify $\frac{- 144 \pm 8 \sqrt{484 - h}}{- 32}$ .

$t = \frac{18 \pm \sqrt{484 - h}}{4}$

Change the $\pm$ to $+$ .

$t = \frac{18 + \sqrt{484 - h}}{4}$

Step 7

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

Simplify the numerator.

Raise $144$ to the power of $2$ .

$t = \frac{- 144 \pm \sqrt{20736 - 4 \cdot - 16 \cdot (160 - h)}}{2 \cdot - 16}$

Multiply $- 4$ by $- 16$ .

$t = \frac{- 144 \pm \sqrt{20736 + 64 \cdot (160 - h)}}{2 \cdot - 16}$

Apply the distributive property.

$t = \frac{- 144 \pm \sqrt{20736 + 64 \cdot 160 + 64 (- h)}}{2 \cdot - 16}$

Multiply $64$ by $160$ .

$t = \frac{- 144 \pm \sqrt{20736 + 10240 + 64 (- h)}}{2 \cdot - 16}$

Multiply $- 1$ by $64$ .

$t = \frac{- 144 \pm \sqrt{20736 + 10240 - 64 h}}{2 \cdot - 16}$

Add $20736$ and $10240$ .

$t = \frac{- 144 \pm \sqrt{30976 - 64 h}}{2 \cdot - 16}$

Factor $64$ out of $30976 - 64 h$ .

Factor $64$ out of $30976$ .

$t = \frac{- 144 \pm \sqrt{64 (484) - 64 h}}{2 \cdot - 16}$

Factor $64$ out of $- 64 h$ .

$t = \frac{- 144 \pm \sqrt{64 (484) + 64 (- h)}}{2 \cdot - 16}$

Factor $64$ out of $64 (484) + 64 (- h)$ .

$t = \frac{- 144 \pm \sqrt{64 (484 - h)}}{2 \cdot - 16}$

Rewrite $64 (484 - h)$ as $8^{2} (22^{2} - h)$ .

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

$t = \frac{- 144 \pm \sqrt{8^{2} (484 - h)}}{2 \cdot - 16}$

Rewrite $484$ as $22^{2}$ .

$t = \frac{- 144 \pm \sqrt{8^{2} (22^{2} - h)}}{2 \cdot - 16}$

Pull terms out from under the radical.

$t = \frac{- 144 \pm 8 \sqrt{22^{2} - h}}{2 \cdot - 16}$

Raise $22$ to the power of $2$ .

$t = \frac{- 144 \pm 8 \sqrt{484 - h}}{2 \cdot - 16}$

Multiply $2$ by $- 16$ .

$t = \frac{- 144 \pm 8 \sqrt{484 - h}}{- 32}$

Simplify $\frac{- 144 \pm 8 \sqrt{484 - h}}{- 32}$ .

$t = \frac{18 \pm \sqrt{484 - h}}{4}$

Change the $\pm$ to $-$ .

$t = \frac{18 - \sqrt{484 - h}}{4}$

Step 8

The final answer is the combination of both solutions.

$t = \frac{18 + \sqrt{484 - h}}{4}$

$t = \frac{18 - \sqrt{484 - h}}{4}$

Write as a Function of h h=-16t^2+144t+160

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