Write as a Function of h h=-4.9t^2+25t

$h = - 4.9 t^{2} + 25 t$

Step 1

Rewrite the equation as $- 4.9 t^{2} + 25 t = h$ .

$- 4.9 t^{2} + 25 t = h$

Step 2

Subtract $h$ from both sides of the equation.

$- 4.9 t^{2} + 25 t - h = 0$

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 $a = - 4.9$ , $b = 25$ , and $c = - h$ into the quadratic formula and solve for $t$ .

$\frac{- 25 \pm \sqrt{25^{2} - 4 \cdot (- 4.9 \cdot (- h))}}{2 \cdot - 4.9}$

Step 5

Simplify.

Simplify the numerator.

Raise $25$ to the power of $2$ .

$t = \frac{- 25 \pm \sqrt{625 - 4 \cdot - 4.9 \cdot (- 1 h)}}{2 \cdot - 4.9}$

Multiply $- 4 \cdot - 4.9 \cdot - 1$ .

Multiply $- 4$ by $- 4.9$ .

$t = \frac{- 25 \pm \sqrt{625 + 19.6 \cdot (- 1 h)}}{2 \cdot - 4.9}$

Multiply $19.6$ by $- 1$ .

$t = \frac{- 25 \pm \sqrt{625 - 19.6 h}}{2 \cdot - 4.9}$

Factor $0.2$ out of $625 - 19.6 h$ .

Factor $0.2$ out of $625$ .

$t = \frac{- 25 \pm \sqrt{0.2 (3125) - 19.6 h}}{2 \cdot - 4.9}$

Factor $0.2$ out of $- 19.6 h$ .

$t = \frac{- 25 \pm \sqrt{0.2 (3125) + 0.2 (- 98 h)}}{2 \cdot - 4.9}$

Factor $0.2$ out of $0.2 (3125) + 0.2 (- 98 h)$ .

$t = \frac{- 25 \pm \sqrt{0.2 (3125 - 98 h)}}{2 \cdot - 4.9}$

Rewrite $0.2 (3125 - 98 h)$ as ${({0.6687403}^{2})}^{2} (3125 - 98 h)$ .

Rewrite $0.2$ as ${0.44721359}^{2}$ .

$t = \frac{- 25 \pm \sqrt{{0.44721359}^{2} (3125 - 98 h)}}{2 \cdot - 4.9}$

Rewrite $0.44721359$ as ${0.6687403}^{2}$ .

$t = \frac{- 25 \pm \sqrt{{({0.6687403}^{2})}^{2} (3125 - 98 h)}}{2 \cdot - 4.9}$

Pull terms out from under the radical.

$t = \frac{- 25 \pm {0.6687403}^{2} \sqrt{3125 - 98 h}}{2 \cdot - 4.9}$

Raise $0.6687403$ to the power of $2$ .

$t = \frac{- 25 \pm 0.44721359 \sqrt{3125 - 98 h}}{2 \cdot - 4.9}$

Multiply $2$ by $- 4.9$ .

$t = \frac{- 25 \pm 0.44721359 \sqrt{3125 - 98 h}}{- 9.8}$

Simplify $\frac{- 25 \pm 0.44721359 \sqrt{3125 - 98 h}}{- 9.8}$ .

$t = \frac{125 \pm 2.23606797 \sqrt{3125 - 98 h}}{49}$

Step 6

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

Simplify the numerator.

Raise $25$ to the power of $2$ .

$t = \frac{- 25 \pm \sqrt{625 - 4 \cdot - 4.9 \cdot (- 1 h)}}{2 \cdot - 4.9}$

Multiply $- 4 \cdot - 4.9 \cdot - 1$ .

Multiply $- 4$ by $- 4.9$ .

$t = \frac{- 25 \pm \sqrt{625 + 19.6 \cdot (- 1 h)}}{2 \cdot - 4.9}$

Multiply $19.6$ by $- 1$ .

$t = \frac{- 25 \pm \sqrt{625 - 19.6 h}}{2 \cdot - 4.9}$

Factor $0.2$ out of $625 - 19.6 h$ .

Factor $0.2$ out of $625$ .

$t = \frac{- 25 \pm \sqrt{0.2 (3125) - 19.6 h}}{2 \cdot - 4.9}$

Factor $0.2$ out of $- 19.6 h$ .

$t = \frac{- 25 \pm \sqrt{0.2 (3125) + 0.2 (- 98 h)}}{2 \cdot - 4.9}$

Factor $0.2$ out of $0.2 (3125) + 0.2 (- 98 h)$ .

$t = \frac{- 25 \pm \sqrt{0.2 (3125 - 98 h)}}{2 \cdot - 4.9}$

Rewrite $0.2 (3125 - 98 h)$ as ${({0.6687403}^{2})}^{2} (3125 - 98 h)$ .

Rewrite $0.2$ as ${0.44721359}^{2}$ .

$t = \frac{- 25 \pm \sqrt{{0.44721359}^{2} (3125 - 98 h)}}{2 \cdot - 4.9}$

Rewrite $0.44721359$ as ${0.6687403}^{2}$ .

$t = \frac{- 25 \pm \sqrt{{({0.6687403}^{2})}^{2} (3125 - 98 h)}}{2 \cdot - 4.9}$

Pull terms out from under the radical.

$t = \frac{- 25 \pm {0.6687403}^{2} \sqrt{3125 - 98 h}}{2 \cdot - 4.9}$

Raise $0.6687403$ to the power of $2$ .

$t = \frac{- 25 \pm 0.44721359 \sqrt{3125 - 98 h}}{2 \cdot - 4.9}$

Multiply $2$ by $- 4.9$ .

$t = \frac{- 25 \pm 0.44721359 \sqrt{3125 - 98 h}}{- 9.8}$

Simplify $\frac{- 25 \pm 0.44721359 \sqrt{3125 - 98 h}}{- 9.8}$ .

$t = \frac{125 \pm 2.23606797 \sqrt{3125 - 98 h}}{49}$

Change the $\pm$ to $+$ .

$t = \frac{125 + 2.23606797 \sqrt{3125 - 98 h}}{49}$

Step 7

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

Simplify the numerator.

Raise $25$ to the power of $2$ .

$t = \frac{- 25 \pm \sqrt{625 - 4 \cdot - 4.9 \cdot (- 1 h)}}{2 \cdot - 4.9}$

Multiply $- 4 \cdot - 4.9 \cdot - 1$ .

Multiply $- 4$ by $- 4.9$ .

$t = \frac{- 25 \pm \sqrt{625 + 19.6 \cdot (- 1 h)}}{2 \cdot - 4.9}$

Multiply $19.6$ by $- 1$ .

$t = \frac{- 25 \pm \sqrt{625 - 19.6 h}}{2 \cdot - 4.9}$

Factor $0.2$ out of $625 - 19.6 h$ .

Factor $0.2$ out of $625$ .

$t = \frac{- 25 \pm \sqrt{0.2 (3125) - 19.6 h}}{2 \cdot - 4.9}$

Factor $0.2$ out of $- 19.6 h$ .

$t = \frac{- 25 \pm \sqrt{0.2 (3125) + 0.2 (- 98 h)}}{2 \cdot - 4.9}$

Factor $0.2$ out of $0.2 (3125) + 0.2 (- 98 h)$ .

$t = \frac{- 25 \pm \sqrt{0.2 (3125 - 98 h)}}{2 \cdot - 4.9}$

Rewrite $0.2 (3125 - 98 h)$ as ${({0.6687403}^{2})}^{2} (3125 - 98 h)$ .

Rewrite $0.2$ as ${0.44721359}^{2}$ .

$t = \frac{- 25 \pm \sqrt{{0.44721359}^{2} (3125 - 98 h)}}{2 \cdot - 4.9}$

Rewrite $0.44721359$ as ${0.6687403}^{2}$ .

$t = \frac{- 25 \pm \sqrt{{({0.6687403}^{2})}^{2} (3125 - 98 h)}}{2 \cdot - 4.9}$

Pull terms out from under the radical.

$t = \frac{- 25 \pm {0.6687403}^{2} \sqrt{3125 - 98 h}}{2 \cdot - 4.9}$

Raise $0.6687403$ to the power of $2$ .

$t = \frac{- 25 \pm 0.44721359 \sqrt{3125 - 98 h}}{2 \cdot - 4.9}$

Multiply $2$ by $- 4.9$ .

$t = \frac{- 25 \pm 0.44721359 \sqrt{3125 - 98 h}}{- 9.8}$

Simplify $\frac{- 25 \pm 0.44721359 \sqrt{3125 - 98 h}}{- 9.8}$ .

$t = \frac{125 \pm 2.23606797 \sqrt{3125 - 98 h}}{49}$

Change the $\pm$ to $-$ .

$t = \frac{125 - 2.23606797 \sqrt{3125 - 98 h}}{49}$

Step 8

The final answer is the combination of both solutions.

$t = \frac{125 + 2.23606797 \sqrt{3125 - 98 h}}{49}$

$t = \frac{125 - 2.23606797 \sqrt{3125 - 98 h}}{49}$

Write as a Function of h h=-4.9t^2+25t

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