A group of students dropped a ball from the top of a 409 foot high building and modeled its height as a function of time using the function H(t) = 400 - 16t^2

Question

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

Let's go through each part of your question step by step:

Part A: What does H(0) mean in the given 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. In this case, H(0) = 400 - 16(0)^2 = 400 feet. This means that at the very beginning, the ball is at a height of 400 feet (the height of the building).

Part B: Interpret the value of H(-3) and its reasonableness

H(-3) represents the height of the ball at t = -3 seconds, which would be three seconds before the ball was dropped. When you calculate H(-3):

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

In this context, interpreting H(-3) is not reasonable because negative time does not have a physical meaning in this experiment. It implies a time before the ball was dropped, which isn't applicable in this scenario.

Part C: What is the domain given the context of this problem?

The domain of the function is governed by the time for which we are considering the ball's height. It starts at the moment the ball is dropped (t = 0) and continues until it reaches the ground. The height of the ball will be 0 when it hits the ground, so we first find when H(t) equals 0:

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

Thus, the domain of the function H(t) is \( t \in [0, 5] \).

Part D: What do the x-axis and y-axis represent?

• The x-axis represents time in seconds (t).
• The y-axis represents the height of the ball in feet (H(t)).

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 can be identified because it includes a term with \( t^2 \). Quadratic functions are characterized by their form \( ax^2 + bx + c \), where \( a, b, \) and \( c \) are constants. In this case, \( a = -16 \), \( b = 0 \), and \( c = 400 \). The presence of the squared term indicates that the graph of the function is a parabola, which opens downwards (since the coefficient of \( t^2 \) is negative). This shape is characteristic of the motion of an object under the influence of gravity.

By analyzing these details, we can see how they fit together to describe the scenario of a ball being dropped from a height.