Need help solving problem

Question

A fireworks shell is shot straight up with an initial velocity of 120 feet per second. Its height s in feet after t seconds is approximated by the equation s=120t-16t^2. If the shell is designed to explode when it reaches its maximum height, how long after being fired, and at what height, will the fireworks appear in the sky?

Answer 1

You need to find the maximum height of the parabola (y-value) , and the time (x value) at that maximum height

 

 

You're given a quadratic function: s= -16t^2 + 120t  

 

 

First things first: s is the y-value, the height in feet, and t is the x-value, the time in seconds.  You can use the vertex formula to figure out the the x-value at the vertex (the highest point in your parabola).

 

 --> x = -b/2a

 

 

Looking back at your original quadratic function, a = -16, b = 120 and c = 0   (the coefficient in front of the t^2 is "a", the coefficient in front of t is "b")

 

Plugging in those values, we get:

 

x = -120/2 (-16)

 

x = 3.75

 

 

 

This means that at the vertex, the x-value is 3.75, or in context, 3.75 seconds.

 

Plug this back into the function to find the y-value, the height:

 

s = 120 (3.75) - 16 (3.75)^2

 

s = 450 - 225

 

s = 225

 

So, the y-value at the vertex is 225 feet.

 

Therefore, it'll take 3.75 seconds to reach the vertex, or maximum height, and it'll go 225 feet high.