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.

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.

Answer 1

Sure! Let’s break down each part of the problem related to the height function \( H(t) = 400 - 16t^2 \).

### Part A: What does \( H(0) \) mean in the given context?
In this context, \( H(0) \) represents the height of the ball at time \( t = 0 \) seconds, which is the moment the ball is dropped from the top of the building. Calculating \( H(0) \):

\[
H(0) = 400 - 16(0)^2 = 400 \text{ feet}
\]

Thus, \( H(0) = 400 \) feet means that the ball starts at a height of 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.
Using the function to calculate \( H(-3) \):

\[
H(-3) = 400 - 16(-3)^2 = 400 - 16(9) = 400 - 144 = 256 \text{ feet}
\]

However, interpreting \( H(-3) \) is not reasonable in this context because the negative value of \( t = -3 \) seconds suggests a time before the ball was dropped. Since the scenario only involves the ball's height during and after the drop, negative time does not have a meaningful real-world interpretation in this scenario.

### Part C: What is the domain given the context of this problem?
In the context of this problem, the domain refers to the possible values of time \( t \). The ball can only be dropped at \( t = 0 \) seconds and can fall for positive durations until it hits the ground. To find when the ball hits the ground, we set \( H(t) = 0 \):

\[
0 = 400 - 16t^2 \\
16t^2 = 400 \\
t^2 = 25 \\
t = 5 \text{ seconds}
\]

Thus, the domain of the function is \( [0, 5] \), indicating that \( t \) can take any value from 0 to 5 seconds (inclusive).

### 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 since the ball was dropped.
- The **y-axis** represents the height \( H(t) \) of the ball in feet above the ground.

### 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 it has the form \( H(t) = a t^2 + b t + c \), where \( a \) is a non-zero coefficient of \( t^2 \) (in this case, \( -16\)). The graph of a quadratic function is a parabola. In this specific case, since the leading coefficient is negative, the parabola opens downward, indicating that the height decreases over time until it reaches the ground.