Question
Alicia watched a drone take off from a bridge. The height of the drone (in meters above the ground)
\[t\] 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
\[t\] 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
Answers
GPT-4o mini
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.
### 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.