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NASA launches a rocket at 
t=0 seconds. Its height, in meters above sea-level, as a function of time is given by 
h(t)=-4.9t^(2)+223 t+241.
Assuming that the rocket will splash down into the ocean, at what time does splashdown occur?
The rocket splashes down after 
◻ seconds.
How high above sea-level does the rocket get at its peak?
The rocket peaks at 
◻ meters above sea-level.

NASA launches a rocket at t=0 t=0 seconds. Its height, in meters above sea-level, as a function of time is given by h(t)=4.9t2+223t+241 h(t)=-4.9 t^{2}+223 t+241 .\newlineAssuming that the rocket will splash down into the ocean, at what time does splashdown occur?\newlineThe rocket splashes down after \square seconds.\newlineHow high above sea-level does the rocket get at its peak?\newlineThe rocket peaks at \square meters above sea-level.

Full solution

Q. NASA launches a rocket at t=0 t=0 seconds. Its height, in meters above sea-level, as a function of time is given by h(t)=4.9t2+223t+241 h(t)=-4.9 t^{2}+223 t+241 .\newlineAssuming that the rocket will splash down into the ocean, at what time does splashdown occur?\newlineThe rocket splashes down after \square seconds.\newlineHow high above sea-level does the rocket get at its peak?\newlineThe rocket peaks at \square meters above sea-level.
  1. Identify Equation: Identify the equation for the height of the rocket as a function of time: h(t)=4.9t2+223t+241h(t) = -4.9t^2 + 223t + 241. To find the splashdown time, solve for tt when h(t)=0h(t) = 0.
  2. Calculate Discriminant and Roots: Calculate the discriminant b24acb^2 - 4ac and then the roots.
  3. Calculate Positive Root: Calculate the positive root since time cannot be negative.
  4. Find Peak Height Time: To find the peak height, use the vertex formula t=b2at = -\frac{b}{2a} for the time at which the peak occurs.
  5. Substitute for Peak Height: Substitute tpeakt_{\text{peak}} back into the height equation to find the peak height.

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