Question

a model rocket is launched with an initial upward velocity of 45 m/s. the rocket’s height h (in meters) after t seconds is given by the following
h = 45t - 5t²
find all values of t for which the rocket’s height is 20 meters.
round your answer(s) to the nearest hundredth.
(if there is more than one answer, use the “or” button.)

t = square seconds

Explanation:

Step1: Substitute h=20 into formula

$20 = 45t - 5t^2$

Step2: Rearrange to standard quadratic form

$5t^2 - 45t + 20 = 0$
Simplify by dividing by 5: $t^2 - 9t + 4 = 0$

Step3: Apply quadratic formula $t=\frac{-b\pm\sqrt{b^2-4ac}}{2a}$

Here $a=1$, $b=-9$, $c=4$
$t=\frac{9\pm\sqrt{(-9)^2-4(1)(4)}}{2(1)}=\frac{9\pm\sqrt{81-16}}{2}=\frac{9\pm\sqrt{65}}{2}$

Step4: Calculate and round values

$\sqrt{65}\approx8.0623$
First solution: $t=\frac{9+8.0623}{2}=\frac{17.0623}{2}\approx8.53$
Second solution: $t=\frac{9-8.0623}{2}=\frac{0.9377}{2}\approx0.47$

Answer:

$t=0.47$ seconds or $t=8.53$ seconds