Justin and Elena each launched a toy rocket into the air. The height of Justin's rocket is modeled by the equation [tex]h = -16t^2 + 60t + 2[/tex]. Elena launched her rocket from the same position, but with an initial velocity double that of Justin's. Which equation best models the height of Elena's rocket?

A. [tex]h = -16t^2 + 60t + 4[/tex]
B. [tex]h = -32t^2 + 120t + 4[/tex]
C. [tex]h = -32t^2 + 60t + 2[/tex]
D. [tex]h = -16t^2 + 120t + 2[/tex]



Answer :

To determine the equation for the height of Elena's rocket, we need to consider a few key aspects of the problem: the initial height, the initial velocity, and how gravity affects the rocket's height over time.

1. Initial Height ([tex]\(h_0\)[/tex]): Both Justin and Elena launched their rockets from the same position, which is 2 feet above the ground. Therefore, the initial height for both rockets is [tex]\( h_0 = 2 \)[/tex] feet.

2. Initial Velocity (v): Justin's rocket has an initial velocity of 60 feet per second. Elena's rocket has an initial velocity that is double that of Justin's. Therefore, the initial velocity for Elena's rocket is:
[tex]\[ v = 2 \times 60 = 120 \text{ feet per second} \][/tex]

3. Gravity's Effect ([tex]\(a\)[/tex]): Gravity affects both rockets equally and is given by [tex]\(-16t^2\)[/tex] (where [tex]\(t\)[/tex] is the time in seconds).

Putting all this together, we form the equation for the height of Elena's rocket [tex]\(h(t)\)[/tex]:

[tex]\[ h(t) = -16t^2 + 120t + 2 \][/tex]

Among the provided options:

- [tex]\( h = -16 t^2 + 60 t + 4 \)[/tex]
- [tex]\( h = -32 t^2 + 120 t + 4 \)[/tex]
- [tex]\( h = -32 t^2 + 60 t + 2 \)[/tex]
- [tex]\( h = -16 t^2 + 120 t + 2 \)[/tex]

We compare our derived equation [tex]\( h(t) = -16t^2 + 120t + 2 \)[/tex] to the provided options. The correct equation that matches our derived function for the height of Elena's rocket is:

[tex]\[ h = -16 t^2 + 120 t + 2 \][/tex]