Trigonometry Examples

$85 = 175 t - 16 t^{2}$

Step 1

Rewrite the equation as $175 t - 16 t^{2} = 85$ .

$175 t - 16 t^{2} = 85$

Step 2

Subtract $85$ from both sides of the equation.

$175 t - 16 t^{2} - 85 = 0$

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 $a = - 16$ , $b = 175$ , and $c = - 85$ into the quadratic formula and solve for $t$ .

$\frac{- 175 \pm \sqrt{175^{2} - 4 \cdot (- 16 \cdot - 85)}}{2 \cdot - 16}$

Step 5

Simplify.

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Step 5.1

Simplify the numerator.

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Step 5.1.1

Raise $175$ to the power of $2$ .

$t = \frac{- 175 \pm \sqrt{30625 - 4 \cdot - 16 \cdot - 85}}{2 \cdot - 16}$

Step 5.1.2

Multiply $- 4 \cdot - 16 \cdot - 85$ .

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Step 5.1.2.1

Multiply $- 4$ by $- 16$ .

$t = \frac{- 175 \pm \sqrt{30625 + 64 \cdot - 85}}{2 \cdot - 16}$

Step 5.1.2.2

Multiply $64$ by $- 85$ .

$t = \frac{- 175 \pm \sqrt{30625 - 5440}}{2 \cdot - 16}$

Step 5.1.3

Subtract $5440$ from $30625$ .

$t = \frac{- 175 \pm \sqrt{25185}}{2 \cdot - 16}$

Step 5.2

Multiply $2$ by $- 16$ .

$t = \frac{- 175 \pm \sqrt{25185}}{- 32}$

Step 5.3

Simplify $\frac{- 175 \pm \sqrt{25185}}{- 32}$ .

$t = \frac{175 \pm \sqrt{25185}}{32}$

Step 6

The final answer is the combination of both solutions.

$t = \frac{175 + \sqrt{25185}}{32}, \frac{175 - \sqrt{25185}}{32}$

Step 7

The result can be shown in multiple forms.

Exact Form:

$t = \frac{175 + \sqrt{25185}}{32}, \frac{175 - \sqrt{25185}}{32}$

Decimal Form:

$t = 10.42805706 \dots, 0.50944293 \dots$