Question

a model rocket is launched with an initial upward velocity of 60 m/s. the rockets height h (in meters) after t seconds is given by the following.

$h = 60t-5t^{2}$

find all values of t for which the rockets height is 27 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 $h = 27$ in the equation $h=60t - 5t^{2}$, so we get $27=60t - 5t^{2}$. Rearrange it to the standard - form of a quadratic equation $5t^{2}-60t + 27 = 0$.

Step2: Identify coefficients

For the quadratic equation $ax^{2}+bx + c = 0$ (here $x=t$, $a = 5$, $b=-60$, $c = 27$), use the quadratic formula $t=\frac{-b\pm\sqrt{b^{2}-4ac}}{2a}$.

Step3: Calculate the discriminant

First, calculate the discriminant $\Delta=b^{2}-4ac=(-60)^{2}-4\times5\times27=3600 - 540 = 3060$.

Step4: Find the values of t

$t=\frac{60\pm\sqrt{3060}}{10}=\frac{60\pm\sqrt{36\times85}}{10}=\frac{60\pm6\sqrt{85}}{10}=\frac{30\pm3\sqrt{85}}{5}$.
$t_1=\frac{30 + 3\sqrt{85}}{5}\approx\frac{30+3\times9.22}{5}=\frac{30 + 27.66}{5}=\frac{57.66}{5}=11.532\approx11.53$.
$t_2=\frac{30 - 3\sqrt{85}}{5}\approx\frac{30-27.66}{5}=\frac{2.34}{5}=0.468\approx0.47$.

Answer:

$t = 0.47$ or $t = 11.53$