Question
A rocket is launched in the air. Its height in feet is given by h, of, t, equals, minus, 16, t, squared, plus, 120, th(t)=−16t
2
+120t 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 y-coordinate (or h-coordinate) of the vertex is
and represents
2
+120t 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 y-coordinate (or h-coordinate) of the vertex is
and represents
Answers
GPT-4o mini
The height of the rocket as a function of time is given by the equation:
\[ h(t) = -16t^2 + 120t \]
This is a quadratic function that opens downward (since the coefficient of \( t^2 \) is negative), meaning it will have a maximum point, which is at the vertex.
To find the vertex of the parabola, we use the formula for the x-coordinate (or t-coordinate) of the vertex, \( t = -\frac{b}{2a} \), where \( a \) and \( b \) are the coefficients in the quadratic equation in the standard form \( ax^2 + bx + c \). For our equation, \( a = -16 \) and \( b = 120 \).
Calculating the t-coordinate of the vertex:
\[ t = -\frac{120}{2(-16)} = -\frac{120}{-32} = \frac{120}{32} = 3.75 \]
Now, we can find the corresponding height (y-coordinate or h-coordinate) at \( t = 3.75 \):
\[ h(3.75) = -16(3.75)^2 + 120(3.75) \]
\[ h(3.75) = -16(14.0625) + 450 \]
\[ h(3.75) = -225 + 450 = 225 \]
Now we have the coordinates of the vertex:
- The x-coordinate (or t-coordinate) of the vertex is **3.75 seconds**, and it represents **the time at which the rocket reaches its maximum height**.
- The y-coordinate (or h-coordinate) of the vertex is **225 feet**, and it represents **the maximum height that the rocket reaches during its flight**.
In summary:
- The x-coordinate (or t-coordinate) of the vertex is **3.75 seconds** and represents **the time at which the rocket reaches its maximum height**.
- The y-coordinate (or h-coordinate) of the vertex is **225 feet** and represents **the maximum height that the rocket reaches during its flight**.
\[ h(t) = -16t^2 + 120t \]
This is a quadratic function that opens downward (since the coefficient of \( t^2 \) is negative), meaning it will have a maximum point, which is at the vertex.
To find the vertex of the parabola, we use the formula for the x-coordinate (or t-coordinate) of the vertex, \( t = -\frac{b}{2a} \), where \( a \) and \( b \) are the coefficients in the quadratic equation in the standard form \( ax^2 + bx + c \). For our equation, \( a = -16 \) and \( b = 120 \).
Calculating the t-coordinate of the vertex:
\[ t = -\frac{120}{2(-16)} = -\frac{120}{-32} = \frac{120}{32} = 3.75 \]
Now, we can find the corresponding height (y-coordinate or h-coordinate) at \( t = 3.75 \):
\[ h(3.75) = -16(3.75)^2 + 120(3.75) \]
\[ h(3.75) = -16(14.0625) + 450 \]
\[ h(3.75) = -225 + 450 = 225 \]
Now we have the coordinates of the vertex:
- The x-coordinate (or t-coordinate) of the vertex is **3.75 seconds**, and it represents **the time at which the rocket reaches its maximum height**.
- The y-coordinate (or h-coordinate) of the vertex is **225 feet**, and it represents **the maximum height that the rocket reaches during its flight**.
In summary:
- The x-coordinate (or t-coordinate) of the vertex is **3.75 seconds** and represents **the time at which the rocket reaches its maximum height**.
- The y-coordinate (or h-coordinate) of the vertex is **225 feet** and represents **the maximum height that the rocket reaches during its flight**.