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A toy rocket is launched straight up from ground level with an initial velocity of 144 ft/sec. The height function is given below:

[tex]h(t) = -16t^2 + 144t[/tex]

a) When will the rocket hit the ground?

Answer :

To find out when the toy rocket will hit the ground, we need to determine when its height, [tex]\( h(t) \)[/tex], becomes zero. The height function is given by:

[tex]\[ h(t) = -16t^2 + 144t \][/tex]

Let's solve the equation [tex]\( h(t) = 0 \)[/tex]:

1. Set the height equation to zero:

[tex]\[
-16t^2 + 144t = 0
\][/tex]

2. Factor the equation:

Notice that we can factor out [tex]\( t \)[/tex] from the equation:

[tex]\[
t(-16t + 144) = 0
\][/tex]

3. Find the solutions for [tex]\( t \)[/tex]:

The factored equation [tex]\( t(-16t + 144) = 0 \)[/tex] gives us two solutions when either factor equals zero.

- [tex]\( t = 0 \)[/tex]: This represents the initial time when the rocket is launched.
- [tex]\( -16t + 144 = 0 \)[/tex]: Solve this equation for the other time when the rocket hits the ground.

4. Solve [tex]\( -16t + 144 = 0 \)[/tex] for [tex]\( t \)[/tex]:

[tex]\[
-16t + 144 = 0
\][/tex]

[tex]\[
-16t = -144
\][/tex]

[tex]\[
t = \frac{144}{16}
\][/tex]

[tex]\[
t = 9
\][/tex]

Therefore, the toy rocket will hit the ground at [tex]\( t = 9 \)[/tex] seconds after it is launched.

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