Question

a model rocket is launched with an initial upward velocity of 73 m/s. the rockets height h (in meters) after t seconds is given by the following.
h = 73t - 5t²
find all values of t for which the rockets height is 47 meters.
round your answer(s) to the nearest hundredth.
(if there is more than one answer, use the \or\ button.)

Explanation:

Step1: Set up the equation

Set $h = 47$ in the equation $h=73t - 5t^{2}$, so we get $47=73t - 5t^{2}$. Rearrange it to the standard - form of a quadratic equation $5t^{2}-73t + 47 = 0$.

Step2: Identify coefficients

For the quadratic equation $ax^{2}+bx + c = 0$ (here $x=t$, $a = 5$, $b=-73$, $c = 47$), use the quadratic formula $t=\frac{-b\pm\sqrt{b^{2}-4ac}}{2a}$.

Step3: Calculate the discriminant

First, calculate the discriminant $\Delta=b^{2}-4ac=(-73)^{2}-4\times5\times47=5329 - 940 = 4389$.

Step4: Find the values of t

$t=\frac{73\pm\sqrt{4389}}{10}$. $\sqrt{4389}\approx66.25$. Then $t_1=\frac{73 + 66.25}{10}=\frac{139.25}{10}=13.93$ and $t_2=\frac{73 - 66.25}{10}=\frac{6.75}{10}=0.68$.

Answer:

$t = 0.68$ seconds or $t = 13.93$ seconds