To determine which equation best models the height of Elena's rocket, let's analyze the details given in the problem:
1. Justin's Rocket Equation:
Justin's rocket's height is given by the equation
[tex]\[
h = -16t^2 + 60t + 2
\][/tex]
where [tex]\( h \)[/tex] is the height of the rocket at time [tex]\( t \)[/tex], the coefficient of [tex]\( t^2 \)[/tex] represents the effect of gravity, the coefficient of [tex]\( t \)[/tex] represents the initial velocity, and the constant term represents the initial height.
2. Elena's Initial Velocity:
- The problem states that Elena's rocket is launched with an initial velocity that is double that of Justin's.
- Justin's initial velocity is 60 (the coefficient of [tex]\( t \)[/tex]).
- Therefore, Elena's initial velocity would be [tex]\( 2 \times 60 = 120 \)[/tex].
3. Equation Components:
- The coefficient of [tex]\( t^2 \)[/tex] (the effect of gravity) should remain the same at -16.
- The new initial velocity (coefficient of [tex]\( t \)[/tex]) should be 120.
- The initial height remains unchanged at 2.
Given these components, we can construct the equation for the height of Elena's rocket:
[tex]\[
h = -16t^2 + 120t + 2
\][/tex]
Now, let's match this derived equation with the given options:
- Option 1: [tex]\( h = -16t^2 + 60t + 4 \)[/tex]
- Option 2: [tex]\( h = -32t^2 + 120t + 4 \)[/tex]
- Option 3: [tex]\( h = -32t^2 + 60t + 2 \)[/tex]
- Option 4: [tex]\( h = -16t^2 + 120t + 2 \)[/tex]
The equation that matches our derived one, [tex]\(-16t^2 + 120t + 2\)[/tex], is Option 4.
Thus, the equation that best models the height of Elena's rocket is:
[tex]\[
\boxed{h = -16t^2 + 120t + 2}
\][/tex]
