Question

a model rocket is launched with an initial upward velocity of 188 ft/s. the rockets height h (in feet) after t seconds is given by the following.
( h = 188t - 16t^2 )
find all values of t for which the rockets height is 92 feet.
round your answer(s) to the nearest hundredth.
(if there is more than one answer, use the \or\ button.)
(there is a diagram of a rockets trajectory with height h from the ground and a box to enter t = seconds with a reset and close button.)

Explanation:

Step1: Set height equal to 92

$188t - 16t^2 = 92$

Step2: Rearrange to standard quadratic form

$16t^2 - 188t + 92 = 0$

Step3: Simplify the equation

Divide all terms by 4: $4t^2 - 47t + 23 = 0$

Step4: Apply quadratic formula

For $at^2+bt+c=0$, $t=\frac{-b\pm\sqrt{b^2-4ac}}{2a}$. Here $a=4$, $b=-47$, $c=23$.

$$ t=\frac{47\pm\sqrt{(-47)^2-4\times4\times23}}{2\times4} $$

Step5: Calculate discriminant

$\sqrt{2209 - 368}=\sqrt{1841}\approx42.92$

Step6: Solve for t values

$t_1=\frac{47 + 42.92}{8}=\frac{89.92}{8}\approx11.24$
$t_2=\frac{47 - 42.92}{8}=\frac{4.08}{8}\approx0.51$

Answer:

$t = 0.51$ seconds or $t = 11.24$ seconds