The object takes 12 minutes to land on the ground.
Explanation:
Given that the equation for the object's height s at time t minutes after the launch is [tex]s(t)=-16t^2+144t+576[/tex]
We need to determine the time that it takes for the object to land on the ground.
Time taken:
The time t of an object can be determined by substituting [tex]s(t)=0[/tex]
Thus, we have,
[tex]0=-16t^2+144t+576[/tex]
Switch sides,
Thus, we get,
[tex]-16 t^{2}+144 t+576=0[/tex]
Let us solve the equation using the quadratic formula.
[tex]t=\frac{-144 \pm \sqrt{144^{2}-4(-16) 576}}{2(-16)}[/tex]
[tex]t=\frac{-144 \pm \sqrt{20736+36824}}{-32}[/tex]
[tex]t=\frac{-144 \pm \sqrt{57600}}{-32}[/tex]
[tex]t=\frac{-144 \pm 240}{-32}[/tex]
Thus, the values of t are [tex]t=\frac{-144 + 240}{-32}[/tex] and [tex]t=\frac{-144 -240}{-32}[/tex]
Simplifying, we get,
[tex]t=\frac{96}{-32}[/tex] and [tex]t=\frac{-384}{-32}[/tex]
[tex]t=-3[/tex] and [tex]t=12[/tex]
Since, the value of t cannot be negative, then [tex]t=12[/tex]
Thus, the object takes 12 minutes to land on the ground.