Question
Create a real-world scenario using the function H (t) = -16t^2 + 30t + 2. Make sure to
provide meaning for the y-intercept. (4 points)
provide meaning for the y-intercept. (4 points)
Answers
GPT-4o mini
**Scenario: Throwing a Ball into the Air**
Imagine you are at a community park and you decide to throw a ball straight up into the air for fun. The height of the ball (in feet) after \( t \) seconds can be modeled by the function:
\[ H(t) = -16t^2 + 30t + 2 \]
**Meaning of the Function:**
- \( H(t) \): Height of the ball above the ground in feet at time \( t \) seconds.
- \( -16t^2 \): Represents the gravitational pull on the ball, causing it to decelerate as it rises and accelerating as it falls back down.
- \( 30t \): Represents the initial upward velocity of the ball. In this case, the ball is thrown upward at a speed of 30 feet per second.
- \( 2 \): Represents the initial height from which the ball is thrown, indicating that it is released 2 feet above the ground (perhaps above your waist when standing).
**Y-Intercept Interpretation:**
To find the y-intercept of the function, we set \( t = 0 \):
\[ H(0) = -16(0)^2 + 30(0) + 2 = 2 \]
The y-intercept is \( H(0) = 2 \). This means that at \( t = 0 \) seconds (the moment you throw the ball), the height of the ball is 2 feet above the ground. This initial height is significant because it tells us the starting point of the ball's journey; it was thrown from a position 2 feet above the ground, not from ground level.
**Conclusion:**
As time progresses, the ball will reach a maximum height due to its initial velocity and then descend back down under the force of gravity, all captured by the function \( H(t) \). The scenario encapsulates the physical principles of projectile motion in a relatable context.
Imagine you are at a community park and you decide to throw a ball straight up into the air for fun. The height of the ball (in feet) after \( t \) seconds can be modeled by the function:
\[ H(t) = -16t^2 + 30t + 2 \]
**Meaning of the Function:**
- \( H(t) \): Height of the ball above the ground in feet at time \( t \) seconds.
- \( -16t^2 \): Represents the gravitational pull on the ball, causing it to decelerate as it rises and accelerating as it falls back down.
- \( 30t \): Represents the initial upward velocity of the ball. In this case, the ball is thrown upward at a speed of 30 feet per second.
- \( 2 \): Represents the initial height from which the ball is thrown, indicating that it is released 2 feet above the ground (perhaps above your waist when standing).
**Y-Intercept Interpretation:**
To find the y-intercept of the function, we set \( t = 0 \):
\[ H(0) = -16(0)^2 + 30(0) + 2 = 2 \]
The y-intercept is \( H(0) = 2 \). This means that at \( t = 0 \) seconds (the moment you throw the ball), the height of the ball is 2 feet above the ground. This initial height is significant because it tells us the starting point of the ball's journey; it was thrown from a position 2 feet above the ground, not from ground level.
**Conclusion:**
As time progresses, the ball will reach a maximum height due to its initial velocity and then descend back down under the force of gravity, all captured by the function \( H(t) \). The scenario encapsulates the physical principles of projectile motion in a relatable context.