Since we're observing the vertical motion of the balloon that's ascending, we can use the equation as shown below.
[tex] d = vt + \frac{1}{2}at^{2} [/tex]
where d is the height, v is the velocity, a is the acceleration, and t is the time.
Given that the balloon reached a height of 81 m at a velocity of 12 m/s². Since a body is dropped and is going downwards, acceleration is equal to g = 10 m/s². So, plugging this into the equation, we have
[tex] 81 = 12t + \frac{1}{2}(-10)t^{2} [/tex]
[tex] 81 = 12t - 5t^{2} [/tex]
[tex] -5t^{2} + 12t -81 = 0 [/tex]
Seeing that we have a quadratic equation, we can apply the quadratic formula to look for its positive root (we neglect the negative root since time can not be negative).
[tex] t = \frac{-(12) - \sqrt{(12)^{2} - 4(-5)(81)}}{2(-5)} [/tex]
[tex] t = 5.4 [/tex]
This means that it will take 5.4 seconds for the body to reach the ground.
Answer: 5.4 seconds
