quadratic functions

Question

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 + 343t + 430. 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.

Answer 1

Answer 2

desmos.com/calculator/y3lrevmhiy

Answer 3

Hi Cassidy S

Splashdown after 71.232 seconds

Maximum height was 6432.5 meters above sea level

Given your Rocket Path as the function

h(t) =-4.9t² + 343t + 430

You can simply line it this up with the Quadratic Equation

h(t) = f(x) = ax² + bx + c

Just so we can follow along algebraically:

let x = t

So for your quadratic the

a = -4.9

b = 343

c = 430

It cannot be factored easily so we can plug the values in to the Quadratic Formula to find the time it takes reach the earth again, the positive x intercept, in this case the ocean surface, Since distance and time are positive only the positive result is useful.

x = t = (-b±SQRT(b²- 4*a*c))/2a

x = t = (-343±SQRT(343²- 4*- 4.9*430))/-9.8

x = t = (-343±SQRT(117649 + 8428))/-9.8

x = t = (-343±SQRT(126077)/-9.8

x = t = (-343±355.073)/-9.8

Since only the positive result applies

x = t = (-343-355.073)/-9.8 = -698.073/-9.8 = 71.232 seconds

Since a is negative the parabola opens downward, so the vertex will be the maximum height the rocket reaches

The x = t coordinate of the vertex is -b/2a = -343/-9.8 = 35 seconds

The f(x) = h(t) coordinate is found by plugging this back into given equation

h(35) = -4.9(35²) + 343(35) + 430 = -6002.5 + 12005 + 430 = 6432.5meters

You can check all the above with a graphing calculator, Desmos.com or some other online graphing calculator. If you are not familiar with Desmos.com it does come with free lessons. Also the answers above were done algebraically just in case you have not started learning derivatives just yet. Considering the (t, h(t)) coordinates, Desmos.com is easier to see than a hand held calculator and derivatives in calculus or pre-calculus are faster and shorter than the algebra above.