Question

a model rocket is launched with an initial upward velocity of 215 ft/s. the rockets height h (in feet) after t seconds is given by the following.
$h = 215t - 16t^2$
find all values of t for which the rockets height is 97 feet.
round your answer(s) to the nearest hundredth.
if there is more than one answer, use the \or\ button.

Explanation:

Step1: Set height equal to 97

$97 = 215t - 16t^2$

Step2: Rearrange to standard quadratic form

$16t^2 - 215t + 97 = 0$

Step3: Apply quadratic formula

For $ax^2+bx+c=0$, $t=\frac{-b\pm\sqrt{b^2-4ac}}{2a}$. Here $a=16$, $b=-215$, $c=97$:
$$t=\frac{215\pm\sqrt{(-215)^2-4(16)(97)}}{2(16)}$$

Step4: Calculate discriminant

$\sqrt{46225 - 6208}=\sqrt{40017}\approx200.04$

Step5: Compute two solutions

$t_1=\frac{215+200.04}{32}\approx\frac{415.04}{32}\approx12.97$
$t_2=\frac{215-200.04}{32}\approx\frac{14.96}{32}\approx0.47$

Answer:

$t=0.47$ seconds or $t=12.97$ seconds