Question
A rocket is launched in the air. Its height in feet is given by h, equals, minus, 16, t, squared, plus, 16, th=−16t
2
+16t where tt represents the time in seconds after launch. Interpret the coordinates of the vertex in context.
Answer
Attempt 1 out of 2
The x-coordinate (or t-coordinate) of the vertex is
and represents
the rocket's height
at its peak
.
The y-coordinate (or h-coordinate) of the vertex is
and represents
how long the rocket has been in the air
when it hits the ground
.
2
+16t where tt represents the time in seconds after launch. Interpret the coordinates of the vertex in context.
Answer
Attempt 1 out of 2
The x-coordinate (or t-coordinate) of the vertex is
and represents
the rocket's height
at its peak
.
The y-coordinate (or h-coordinate) of the vertex is
and represents
how long the rocket has been in the air
when it hits the ground
.
Answers
GPT-4o mini
The height of the rocket is given by the equation \( h(t) = -16t^2 + 16t \). This is a quadratic equation in the form \( h(t) = at^2 + bt + c \), where \( a = -16 \), \( b = 16 \), and \( c = 0 \).
To find the vertex of this parabola, we can use the formula for the x-coordinate (or t-coordinate) of the vertex:
\[
t = -\frac{b}{2a}
\]
Plugging in the values of \( a \) and \( b \):
\[
t = -\frac{16}{2 \times -16} = \frac{16}{32} = 0.5 \text{ seconds}
\]
Next, we substitute \( t = 0.5 \) back into the height equation to find the y-coordinate (or h-coordinate) of the vertex:
\[
h(0.5) = -16(0.5)^2 + 16(0.5)
\]
\[
= -16(0.25) + 8
\]
\[
= -4 + 8
\]
\[
= 4 \text{ feet}
\]
Thus, the coordinates of the vertex are \( (0.5, 4) \).
Now, interpreting these coordinates in context:
- The x-coordinate (or t-coordinate) of the vertex is **0.5**, which represents **the time in seconds after launch when the rocket reaches its peak height**.
- The y-coordinate (or h-coordinate) of the vertex is **4**, which represents **the maximum height of the rocket, which is 4 feet**.
So, the final answer is:
The x-coordinate (or t-coordinate) of the vertex is **0.5 seconds** and represents **the time when the rocket reaches its peak height**. The y-coordinate (or h-coordinate) of the vertex is **4 feet** and represents **the maximum height of the rocket**.
To find the vertex of this parabola, we can use the formula for the x-coordinate (or t-coordinate) of the vertex:
\[
t = -\frac{b}{2a}
\]
Plugging in the values of \( a \) and \( b \):
\[
t = -\frac{16}{2 \times -16} = \frac{16}{32} = 0.5 \text{ seconds}
\]
Next, we substitute \( t = 0.5 \) back into the height equation to find the y-coordinate (or h-coordinate) of the vertex:
\[
h(0.5) = -16(0.5)^2 + 16(0.5)
\]
\[
= -16(0.25) + 8
\]
\[
= -4 + 8
\]
\[
= 4 \text{ feet}
\]
Thus, the coordinates of the vertex are \( (0.5, 4) \).
Now, interpreting these coordinates in context:
- The x-coordinate (or t-coordinate) of the vertex is **0.5**, which represents **the time in seconds after launch when the rocket reaches its peak height**.
- The y-coordinate (or h-coordinate) of the vertex is **4**, which represents **the maximum height of the rocket, which is 4 feet**.
So, the final answer is:
The x-coordinate (or t-coordinate) of the vertex is **0.5 seconds** and represents **the time when the rocket reaches its peak height**. The y-coordinate (or h-coordinate) of the vertex is **4 feet** and represents **the maximum height of the rocket**.