A rocket is launched at t = 0 seconds. Its height, in meters above sea-level, as a function of time is given by h(t) = -4.9t^2 + 292 t + 94.

Question

A rocket is launched at t = 0 seconds. Its height, in meters above sea-level, as a function of time is given by h(t) = -4.9t^2 + 292 t + 94.

At what time does the rocket reach its maximum height? (Round answer to 2 decimal places.)

The rocket reaches its maximum height after_______seconds.

At what height does the rocket reach its maximum height above the water? (Round answer to 2 decimal places.)

The rocket reaches its maximum height at __________meters above sea-level.

Answer 1

There are 2 ways to solve the problem.

Either:

Use the graph function on the graphing calculator and find the max on the parabola,

 

or

 

Derive the function and solve for t=0, then use that time and plug back into the original equation.

h'(t)=-9.8t+292=0

t=29.8 (29.795918....)  this is the max height time (s)

h(29.8)=4444.204        this is the max height (m)

 

fun fact- -9.8 is the acceleration of gravity.

Answer 2

h(t) = -4.9t² + 292t + 94

 

This is a quadratic equation that graphs as a parabola

opening downward.  The maximum height will be at the

vertex of the parabola...  

The '"t" value at the vertex is -b/2a

The maximum height h(t) will be at h(-b/2a)

a = -4.9,  b=292 

 

-b/2a = -292/2(-4.9) = 29.796 seconds

h(29.796) = -4.9(29.796²)+292(29.796)+94 = 4444.20 meters

 

The rocket reaches a maximum height of 4444.20 meters at time 29.80 seconds

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