Trigonometry Examples

$h = 105 t - 16 t^{2}$

Step 1

Rewrite the equation as $105 t - 16 t^{2} = h$ .

$105 t - 16 t^{2} = h$

Step 2

Subtract $h$ from both sides of the equation.

$105 t - 16 t^{2} - h = 0$

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 $a = - 16$ , $b = 105$ , and $c = - h$ into the quadratic formula and solve for $t$ .

$\frac{- 105 \pm \sqrt{105^{2} - 4 \cdot (- 16 \cdot (- h))}}{2 \cdot - 16}$

Step 5

Simplify.

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Step 5.1

Simplify the numerator.

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Step 5.1.1

Raise $105$ to the power of $2$ .

$t = \frac{- 105 \pm \sqrt{11025 - 4 \cdot - 16 \cdot (- 1 h)}}{2 \cdot - 16}$

Step 5.1.2

Multiply $- 4 \cdot - 16 \cdot - 1$ .

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Step 5.1.2.1

Multiply $- 4$ by $- 16$ .

$t = \frac{- 105 \pm \sqrt{11025 + 64 \cdot (- 1 h)}}{2 \cdot - 16}$

Step 5.1.2.2

Multiply $64$ by $- 1$ .

$t = \frac{- 105 \pm \sqrt{11025 - 64 h}}{2 \cdot - 16}$

Step 5.2

Multiply $2$ by $- 16$ .

$t = \frac{- 105 \pm \sqrt{11025 - 64 h}}{- 32}$

Step 5.3

Simplify $\frac{- 105 \pm \sqrt{11025 - 64 h}}{- 32}$ .

$t = \frac{105 \pm \sqrt{11025 - 64 h}}{32}$

Step 6

The final answer is the combination of both solutions.

$t = \frac{105 + \sqrt{11025 - 64 h}}{32}$

$t = \frac{105 - \sqrt{11025 - 64 h}}{32}$