a model rocket is launched with an initial upward velocity of 67 m/s. the rockets height h (in meters) after…

a model rocket is launched with an initial upward velocity of 67 m/s. the rockets height h (in meters) after t seconds is given by the following. h = 67t - 5t² find all values of t for which the rockets height is 30 meters. round your answer(s) to the nearest hundredth. (if there is more than one answer, use the \or\ button.)

a model rocket is launched with an initial upward velocity of 67 m/s. the rockets height h (in meters) after t seconds is given by the following. h = 67t - 5t² find all values of t for which the rockets height is 30 meters. round your answer(s) to the nearest hundredth. (if there is more than one answer, use the \or\ button.)

Answer

Explanation:

Step1: Set up the equation

Set $h = 30$ in $h=67t - 5t^{2}$, so we get $30=67t - 5t^{2}$. Rearrange it to the standard - quadratic form $5t^{2}-67t + 30 = 0$.

Step2: Identify coefficients

For the quadratic equation $ax^{2}+bx + c = 0$ (here $x=t$, $a = 5$, $b=-67$, $c = 30$).

Step3: Use the quadratic formula

The quadratic formula is $t=\frac{-b\pm\sqrt{b^{2}-4ac}}{2a}$. Substitute $a = 5$, $b=-67$, $c = 30$ into it: $t=\frac{67\pm\sqrt{(-67)^{2}-4\times5\times30}}{2\times5}=\frac{67\pm\sqrt{4489 - 600}}{10}=\frac{67\pm\sqrt{3889}}{10}$.

Step4: Calculate the values of t

$\sqrt{3889}\approx62.36$. Then $t_1=\frac{67 + 62.36}{10}=\frac{129.36}{10}=12.94$ and $t_2=\frac{67 - 62.36}{10}=\frac{4.64}{10}=0.46$.

Answer:

$t = 0.46$ or $t = 12.94$