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

$h = 73t - 5t^{2}$

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 quadratic - form $5t^{2}-73t + 47 = 0$.

Step2: Identify the 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

Substitute into the quadratic formula:
$t=\frac{73\pm\sqrt{4389}}{10}$.
$\sqrt{4389}\approx66.25$.
$t_1=\frac{73 + 66.25}{10}=\frac{139.25}{10}=13.93$ (rounded to the nearest hundredth).
$t_2=\frac{73 - 66.25}{10}=\frac{6.75}{10}=0.68$ (rounded to the nearest hundredth).

Answer:

$t = 0.68$ or $t = 13.93$