A rocket is launched from a tower. The height of the rocket, y in feet, is related to the time after launch, x in seconds, by the given...?

y = -16x² + 152x + 83

We first find the values of x for which y = 0

i.e. -16x² + 152x + 83 = 0

or, 16x² - 152x - 83 = 0

Using the quadratic formula we get:

x = (152 ± 16√111)/32

i.e. x = (152 - 16√111)/32 and x = (152 + 16√111)/32

As the parabola is ' ∩ ' shaped the maximum value will occur midway between these values

i.e. [(152 - 16√111)/32 + (152 + 16√111)/32]/2

so, (304/32)/2 => 304/64 = 19/4 = 4.75

Therefore, y = -16(4.75)² + 152(4.75) + 83

Hence, y = 444 feet...maximum height.

A sketch is below.

IS THAT ENOUGH WORK FOR YOU??

:)>

See below for work and solution. Without calculus, the graph shows that the highest point occurs at x = 4.75.

When x = 4.75, f(4.75) = 444 so

the maximum height reached by the rocket is 444 ft.......ANS

What you have there is the equation of a parabola (in particular a downward facing parabola because the leading coefficient is negative). That makes sense because the rocket will go up, reach its maximum height and then come back down.

The *vertex* will be the highest point of the parabola.

If you haven't already, you should learn the formula for the x-coordinate of the vertex.

x = -b/(2a)

If you need help remembering that, it's essentially the quadratic formula without the ±√ part.

In your equation:

a = -16

b = 152

x = -152/(2*-16)

x = -152/-32

x = 19/4

x = 4.75

So 4.75 seconds after launch, the rocket has reached its maximum height. Now all you have to do is plug in x = 4.75 and calculate the height (y).

y = -16x² + 152x + 83

y = -16(4.75)² + 152(4.75) + 83

y = 444 feet

Answer:

The maximum height is 444 feet.

The graph below should help you visualize it.

x= 152/32= 4.75

y(4.75)= 444 ft