Question
a player kicks a ball up in the air, the function h(t)= -16t^2 + 40t can be used to find the height, h(t), of the ball above the field in feet, where t is the time in seconds after the ball is kicked, what is the domain of the function for this situation
Answers
GPT-4 Turbo
The domain of the function for this situation is the set of all possible values of \( t \) (time in seconds) for which the function \( h(t) \) gives a meaningful result—which is the height of the ball above the field.
Given the function \( h(t) = -16t^2 + 40t \), the only physical constraint on the domain is that time cannot be negative, and the ball cannot have a negative height—it can't go below the ground level in this context. Therefore, we only consider values of \( t \) where the height is zero or positive. This occurs from the time the ball is kicked (when \( t = 0 \)) until it returns to the ground (when \( h(t) = 0 \)).
To find out when the ball hits the ground, we can set \( h(t) \) to zero and solve for \( t \):
\( 0 = -16t^2 + 40t \)
We can factor out \( t \) from each term:
\( t(-16t + 40) = 0 \)
The equation yields two solutions for \( t \):
\( t = 0 \) (when the ball is kicked), and
\( -16t + 40 = 0 \) so \( t = 40/16 = 2.5 \) seconds (when the ball hits the ground).
Since the ball is only in the air from the time it is kicked until it hits the ground, the domain of the function in this physical context is \( [0, 2.5] \) seconds. Thus, the function h(t) provides valid results for the height of the ball for any time \( t \) within this interval. This is known as a closed interval since it includes the endpoints.
Given the function \( h(t) = -16t^2 + 40t \), the only physical constraint on the domain is that time cannot be negative, and the ball cannot have a negative height—it can't go below the ground level in this context. Therefore, we only consider values of \( t \) where the height is zero or positive. This occurs from the time the ball is kicked (when \( t = 0 \)) until it returns to the ground (when \( h(t) = 0 \)).
To find out when the ball hits the ground, we can set \( h(t) \) to zero and solve for \( t \):
\( 0 = -16t^2 + 40t \)
We can factor out \( t \) from each term:
\( t(-16t + 40) = 0 \)
The equation yields two solutions for \( t \):
\( t = 0 \) (when the ball is kicked), and
\( -16t + 40 = 0 \) so \( t = 40/16 = 2.5 \) seconds (when the ball hits the ground).
Since the ball is only in the air from the time it is kicked until it hits the ground, the domain of the function in this physical context is \( [0, 2.5] \) seconds. Thus, the function h(t) provides valid results for the height of the ball for any time \( t \) within this interval. This is known as a closed interval since it includes the endpoints.