Question
An object is launched at 64 ft per second from an elevated platform. The function f(x)=-16x^2+64x+6 models its trajectory over time, x, in seconds. Which of the following are true?
A: The object reaches its maximum height at 2 seconds.
B: The height of the platform is 6 ft.
C: The maximum height of the object is 70ft.
D: The object will be higher than 60 feet at 1 second.
E: The height of the object increases and then decreases.
A: The object reaches its maximum height at 2 seconds.
B: The height of the platform is 6 ft.
C: The maximum height of the object is 70ft.
D: The object will be higher than 60 feet at 1 second.
E: The height of the object increases and then decreases.
Answers
GPT-4o mini
To analyze the trajectory of the object given by the function \( f(x) = -16x^2 + 64x + 6 \), we can evaluate each statement individually.
**A: The object reaches its maximum height at 2 seconds.**
To find the time at which the maximum height occurs for a quadratic function in the form \( ax^2 + bx + c \), we can use the formula \( x = -\frac{b}{2a} \).
Here, \( a = -16 \) and \( b = 64 \):
\[
x = -\frac{64}{2 \times -16} = -\frac{64}{-32} = 2
\]
Thus, the object does reach its maximum height at 2 seconds. **True**
---
**B: The height of the platform is 6 ft.**
The height of the platform corresponds to the function's value at \( x = 0 \):
\[
f(0) = -16(0)^2 + 64(0) + 6 = 6
\]
Therefore, the height of the platform is indeed 6 ft. **True**
---
**C: The maximum height of the object is 70 ft.**
To find the maximum height, we evaluate the function at \( x = 2 \):
\[
f(2) = -16(2)^2 + 64(2) + 6 = -16(4) + 128 + 6 = -64 + 128 + 6 = 70
\]
Thus, the maximum height of the object is 70 ft. **True**
---
**D: The object will be higher than 60 feet at 1 second.**
We can check the height at \( x = 1 \):
\[
f(1) = -16(1)^2 + 64(1) + 6 = -16 + 64 + 6 = 54
\]
Since 54 ft is less than 60 ft, this statement is **False**.
---
**E: The height of the object increases and then decreases.**
Since the parabola opens downwards (as the coefficient of \( x^2 \) is negative), and we have already established that the object reaches a maximum height, the object's trajectory does indeed increase to that maximum height and then decrease afterward. **True**
---
In summary:
- A: True
- B: True
- C: True
- D: False
- E: True
**A: The object reaches its maximum height at 2 seconds.**
To find the time at which the maximum height occurs for a quadratic function in the form \( ax^2 + bx + c \), we can use the formula \( x = -\frac{b}{2a} \).
Here, \( a = -16 \) and \( b = 64 \):
\[
x = -\frac{64}{2 \times -16} = -\frac{64}{-32} = 2
\]
Thus, the object does reach its maximum height at 2 seconds. **True**
---
**B: The height of the platform is 6 ft.**
The height of the platform corresponds to the function's value at \( x = 0 \):
\[
f(0) = -16(0)^2 + 64(0) + 6 = 6
\]
Therefore, the height of the platform is indeed 6 ft. **True**
---
**C: The maximum height of the object is 70 ft.**
To find the maximum height, we evaluate the function at \( x = 2 \):
\[
f(2) = -16(2)^2 + 64(2) + 6 = -16(4) + 128 + 6 = -64 + 128 + 6 = 70
\]
Thus, the maximum height of the object is 70 ft. **True**
---
**D: The object will be higher than 60 feet at 1 second.**
We can check the height at \( x = 1 \):
\[
f(1) = -16(1)^2 + 64(1) + 6 = -16 + 64 + 6 = 54
\]
Since 54 ft is less than 60 ft, this statement is **False**.
---
**E: The height of the object increases and then decreases.**
Since the parabola opens downwards (as the coefficient of \( x^2 \) is negative), and we have already established that the object reaches a maximum height, the object's trajectory does indeed increase to that maximum height and then decrease afterward. **True**
---
In summary:
- A: True
- B: True
- C: True
- D: False
- E: True