Sure! Let's break down each part of the question:
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 when the ball is released from the top of the building. Since the equation is given as \( H(t) = 400 - 16t^2 \), substituting \( t = 0 \) gives \( H(0) = 400 - 16(0^2) = 400 \). Therefore, \( H(0) = 400 \) means that the ball is at a height of 400 feet, which is the initial height of the building.
Part B:
Explain whether or not interpreting the value of \( H(-3) \) is reasonable in this context.
Interpreting \( H(-3) \) means evaluating the height of the ball at \( -3 \) seconds, which suggests looking at the height 3 seconds before the ball was dropped. In this physical context, negative time does not have a meaningful interpretation, as the ball has not yet been released from the building at that time. Therefore, interpreting \( H(-3) \) is not reasonable in this context, as it does not represent a real event in the timeline of the ball's drop.
Part C:
What is the domain given the context of this problem?
The domain of the function \( H(t) \) represents all the possible values of time \( t \) for which the height \( H(t) \) is meaningful. In the context of this problem, time starts when the ball is dropped (at \( t = 0 \)) and continues until the ball hits the ground. To find the time when the ball hits the ground, we set \( H(t) = 0 \): \[ 400 - 16t^2 = 0 \ 16t^2 = 400 \ t^2 = 25 \ t = 5 \] Thus, \( t \) can range from \( 0 \) to \( 5 \) seconds. Therefore, the domain is \( [0, 5] \).
Part D:
What does the x-axis represent? What does the y-axis represent?
In this context, the x-axis represents time \( t \) measured in seconds, indicating how much time has passed since the ball was dropped from the building. The y-axis represents height \( H(t) \) measured in feet, indicating the height of the ball above the ground at any given time \( t \).
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. We know this because it can be expressed in the standard form of a quadratic equation, \( H(t) = at^2 + bt + c \), where \( a, b, \) and \( c \) are constants, and in this case, \( a = -16 \), \( b = 0 \), and \( c = 400 \). The presence of the \( t^2 \) term indicates that as \( t \) increases, \( H(t) \) changes in a parabolic manner, which characterizes quadratic functions. Additionally, the negative coefficient before the \( t^2 \) term signifies that the parabola opens downward, which is consistent with the height decreasing as time progresses due to the ball falling.