Question
A group of physics students dropped a ball from the top of a 400-foot-high building and modeled its height as a function of time using the function H(t)=400−16t2 . The height, H , is measured in feet, and time, t , is measured in seconds. Desmos Calculator (click here) Answer all parts below. Make sure you clearly label each part as Part A, Part B, Part C, Part D and Part E in your response. Part A: What does H(0) mean in the given context? Part B: Jon is asked to interpret the value of H(−3) . Explain whether or not interpreting this value is reasonable in this context. Part C: What is the domain given the context of this problem? Part D: What does the x-axis represent? What does the y-axis represent? Part E: What kind of function is this (linear, quadratic, exponential)? Explain how you know.
Answers
GPT-4o mini
Sure! Here are the responses to each part you're interested in:
### Part A:
**What does H(0) mean in the given context?**
In the context of the problem, \( H(0) \) represents the height of the ball at time \( t = 0 \) seconds, which is when the ball is initially dropped from the top of the building. According to the function \( H(t) = 400 - 16t^2 \), when \( t = 0 \), we find that \( H(0) = 400 - 16(0)^2 = 400 \) feet. This means that at the moment the ball is dropped, its height is 400 feet, which is the height of the building.
### Part B:
**Jon is asked to interpret the value of H(−3). Explain whether or not interpreting this value is reasonable in this context.**
The value \( H(-3) \) corresponds to the height of the ball at \( t = -3 \) seconds, which indicates a time 3 seconds before the ball was dropped. In a physical context where we are measuring time from the moment the ball is dropped (which is time \( t = 0 \)), interpreting \( H(-3) \) is not reasonable because negative time does not align with the scenario being modeled. We cannot have a physical situation where the ball is in a state before it was dropped, so this interpretation does not have a meaningful context.
### Part C:
**What is the domain given the context of this problem?**
The domain of the function \( H(t) = 400 - 16t^2 \) in this context is restricted to non-negative values of \( t \) since we are measuring time from the moment the ball is dropped. The function is defined as long as \( H(t) \) remains non-negative, which represents the height of the ball. The height becomes 0 when \( 400 - 16t^2 = 0 \), leading to \( t^2 = 25 \) or \( t = 5 \) seconds. Therefore, the domain is \( 0 \leq t \leq 5 \).
### Part D:
**What does the x-axis represent? What does the y-axis represent?**
In this context, the x-axis represents time \( t \) in seconds, which measures how long it has been since the ball was dropped. The y-axis represents the height \( H \) of the ball in feet above the ground as a function of time. Thus, as you move along the x-axis (time), the corresponding value on the y-axis indicates how high the ball is at that moment in time.
### Part E:
**What kind of function is this (linear, quadratic, exponential)? Explain how you know.**
The function \( H(t) = 400 - 16t^2 \) is a **quadratic function**. This is evident because the highest exponent of the variable \( t \) in the equation is 2, which is characteristic of quadratic functions. Quadratic functions typically have the general form \( ax^2 + bx + c \). In this case, the term \( -16t^2 \) indicates that it is a downward-opening parabola since the coefficient of \( t^2 \) is negative. The graph of this function will show a curve that opens downwards and has a vertex at the maximum height of the ball before it begins to fall back down to the ground.
### Part A:
**What does H(0) mean in the given context?**
In the context of the problem, \( H(0) \) represents the height of the ball at time \( t = 0 \) seconds, which is when the ball is initially dropped from the top of the building. According to the function \( H(t) = 400 - 16t^2 \), when \( t = 0 \), we find that \( H(0) = 400 - 16(0)^2 = 400 \) feet. This means that at the moment the ball is dropped, its height is 400 feet, which is the height of the building.
### Part B:
**Jon is asked to interpret the value of H(−3). Explain whether or not interpreting this value is reasonable in this context.**
The value \( H(-3) \) corresponds to the height of the ball at \( t = -3 \) seconds, which indicates a time 3 seconds before the ball was dropped. In a physical context where we are measuring time from the moment the ball is dropped (which is time \( t = 0 \)), interpreting \( H(-3) \) is not reasonable because negative time does not align with the scenario being modeled. We cannot have a physical situation where the ball is in a state before it was dropped, so this interpretation does not have a meaningful context.
### Part C:
**What is the domain given the context of this problem?**
The domain of the function \( H(t) = 400 - 16t^2 \) in this context is restricted to non-negative values of \( t \) since we are measuring time from the moment the ball is dropped. The function is defined as long as \( H(t) \) remains non-negative, which represents the height of the ball. The height becomes 0 when \( 400 - 16t^2 = 0 \), leading to \( t^2 = 25 \) or \( t = 5 \) seconds. Therefore, the domain is \( 0 \leq t \leq 5 \).
### Part D:
**What does the x-axis represent? What does the y-axis represent?**
In this context, the x-axis represents time \( t \) in seconds, which measures how long it has been since the ball was dropped. The y-axis represents the height \( H \) of the ball in feet above the ground as a function of time. Thus, as you move along the x-axis (time), the corresponding value on the y-axis indicates how high the ball is at that moment in time.
### Part E:
**What kind of function is this (linear, quadratic, exponential)? Explain how you know.**
The function \( H(t) = 400 - 16t^2 \) is a **quadratic function**. This is evident because the highest exponent of the variable \( t \) in the equation is 2, which is characteristic of quadratic functions. Quadratic functions typically have the general form \( ax^2 + bx + c \). In this case, the term \( -16t^2 \) indicates that it is a downward-opening parabola since the coefficient of \( t^2 \) is negative. The graph of this function will show a curve that opens downwards and has a vertex at the maximum height of the ball before it begins to fall back down to the ground.