Question
You are standing on a balcony and drop a tennis ball from your hand. The height of the ball, h(x)
, (in feet) above the ground at any time x
(in seconds) is modeled by the function:
h(x)=−16x2+64x+200
where:
x
represents time in seconds after the ball is dropped.
h(x)
represents the height of the ball in feet.
The given function intersects the x-axis at 6.06 and the y-axis at 200 and has a vertex at (2,264). Interpret what each of these values mean in the context of the problem.
, (in feet) above the ground at any time x
(in seconds) is modeled by the function:
h(x)=−16x2+64x+200
where:
x
represents time in seconds after the ball is dropped.
h(x)
represents the height of the ball in feet.
The given function intersects the x-axis at 6.06 and the y-axis at 200 and has a vertex at (2,264). Interpret what each of these values mean in the context of the problem.
Answers
Answered by
GPT-4o mini
The function \( h(x) = -16x^2 + 64x + 200 \) describes the height of a tennis ball dropped from a balcony over time. Let's interpret the values you mentioned in the context of this problem.
1. **Intersection with the y-axis (\( h(0) = 200 \)):**
- When \( x = 0 \), \( h(0) = 200 \). This indicates that at time \( x = 0 \) seconds (the moment the ball is dropped), the height of the ball is 200 feet above the ground. This represents the height of the balcony from which the ball is dropped.
2. **Intersection with the x-axis (\( x = 6.06 \)):**
- The value \( x = 6.06 \) seconds means that the ball reaches a height of \( h(x) = 0 \) (i.e., the ground level) at approximately 6.06 seconds after it is dropped. This indicates that the ball will hit the ground at this time.
3. **Vertex at (2, 264):**
- The vertex of the parabola is located at \( (2, 264) \). This means that 2 seconds after the ball is dropped, it reaches its maximum height of 264 feet. The vertex represents the peak of the ball's motion; after reaching this maximum height, the ball starts to descend.
In summary:
- The height of the ball when dropped is 200 feet.
- The ball takes approximately 6.06 seconds to hit the ground.
- The maximum height the ball reaches during its flight is 264 feet, occurring 2 seconds after it is dropped.
1. **Intersection with the y-axis (\( h(0) = 200 \)):**
- When \( x = 0 \), \( h(0) = 200 \). This indicates that at time \( x = 0 \) seconds (the moment the ball is dropped), the height of the ball is 200 feet above the ground. This represents the height of the balcony from which the ball is dropped.
2. **Intersection with the x-axis (\( x = 6.06 \)):**
- The value \( x = 6.06 \) seconds means that the ball reaches a height of \( h(x) = 0 \) (i.e., the ground level) at approximately 6.06 seconds after it is dropped. This indicates that the ball will hit the ground at this time.
3. **Vertex at (2, 264):**
- The vertex of the parabola is located at \( (2, 264) \). This means that 2 seconds after the ball is dropped, it reaches its maximum height of 264 feet. The vertex represents the peak of the ball's motion; after reaching this maximum height, the ball starts to descend.
In summary:
- The height of the ball when dropped is 200 feet.
- The ball takes approximately 6.06 seconds to hit the ground.
- The maximum height the ball reaches during its flight is 264 feet, occurring 2 seconds after it is dropped.