Alicia watched a drone take off from a bridge. The height of the drone (in meters above the ground)

Question

\[t\] minutes after takeoff is modeled by
\[h(t)=-3t^2+12t+96\]
Alicia wants to know when the drone will land on the ground.
1) Rewrite the function in a different form (factored or vertex) where the answer appears as a number in the equation.
\[h(t)=\]
2) How many minutes after takeoff does the drone land on the ground?
minutes

Answer 1

To solve this problem, we will start by rewriting the function \(h(t) = -3t^2 + 12t + 96\) and then find when the drone lands on the ground by solving \(h(t) = 0\).

1) Rewriting the function

We can factor the equation by first factoring out the coefficient of \(t^2\) from the quadratic.

1. Factor out \(-3\): \[ h(t) = -3(t^2 - 4t - 32) \]

2. Now we will factor the quadratic \(t^2 - 4t - 32\). To factor this, we need to find two numbers that multiply to \(-32\) (the constant term) and add to \(-4\) (the coefficient of \(t\)). The factors are \(-8\) and \(4\).

   So, we can write: \[ t^2 - 4t - 32 = (t - 8)(t + 4) \]

3. Incorporating the factor of \(-3\), we rewrite \(h(t)\): \[ h(t) = -3(t - 8)(t + 4) \]

Thus, the function in its factored form is: \[ h(t) = -3(t - 8)(t + 4) \]

2) Finding when the drone lands on the ground

To determine when the drone lands on the ground, we set \(h(t)\) to 0: \[ -3(t - 8)(t + 4) = 0 \]

This will give us two solutions: \[ (t - 8) = 0 \quad \text{or} \quad (t + 4) = 0 \]

From the first equation: \[ t - 8 = 0 \Rightarrow t = 8 \]

From the second equation: \[ t + 4 = 0 \Rightarrow t = -4 \]

Since time cannot be negative in this context, we discard \(t = -4\).

Thus, the drone lands on the ground \(8\) minutes after takeoff.

Final Answers

1. \[h(t) = -3(t - 8)(t + 4)\]
2. The drone lands on the ground after \(8\) minutes.