Question

model rocket is launched from a raised platform at a speed of 76 feet per second. Its height in feet is given by h(t)=-16t^2+76t+24 (t=seconds after launch) What is the maximum height reached by the rocket? and what time does it meet maximum height? And how long is the rocket in the air?​

Answer 1

Hello!

This is a question relating a quadratic equation to the vertex and roots.

Since this parabola has a negative a value, the vertex will be the maximum height that this rocket reaches.

We can find the vertex (h,k) with the following equations.

$h=\frac{-b}{2a}$

$k=h(h)$

$h=\frac{-76}{2(-16)}$

$h=\frac{76}{32}$

$h=2.375$

$k=-16(2.375)^2+76(2.375)+24$

$k=114.25$

We can interpret the values like this:

At $t=2.375$ seconds after the rocket was launched, the rocket reached its maximum height of $114.25$ feet.

Since the y-intercept is at $(0,24)$ and this is a negative parabola, there will only be one positive root, which will be how long our rocket is in the air.

Use the quadratic formula.

$x=\frac{-b+-\sqrt{b^2-4ac}}{2a}$

$x=\frac{-(76)+-\sqrt{(76)^2-4(-16)(24)}}{2(-16)}$

$x=5.407, -0.297$

Since we are searching for our positive root, this rocket is in the air for 5.407 seconds.

Hope this helps!