Question
The height of a ball thrown upward from the ground is given by the quadratic function
h = 160t - 16
, where h is in feet and t is in seconds.
What is the reasonable domain for this function?
Responses
A {x: 0 ≤ x ≤ 10}{x: 0 ≤ x ≤ 10}
B {x: 0 ≤ x ≤ 160}{x: 0 ≤ x ≤ 160}
C {x: x ≥ 0}{x: x ≥ 0}
D all real numbers
h = 160t - 16
, where h is in feet and t is in seconds.
What is the reasonable domain for this function?
Responses
A {x: 0 ≤ x ≤ 10}{x: 0 ≤ x ≤ 10}
B {x: 0 ≤ x ≤ 160}{x: 0 ≤ x ≤ 160}
C {x: x ≥ 0}{x: x ≥ 0}
D all real numbers
Answers
GPT-4o mini
To determine the reasonable domain for the function \( h(t) = 160t - 16t^2 \), we need to consider the physical context of the problem. The function describes the height of a ball thrown upward, which implies that:
1. Time \( t \) cannot be negative since we start measuring time from the moment the ball is thrown (which is at \( t = 0 \)).
2. We also need to consider when the ball hits the ground (i.e., when \( h = 0 \)), which occurs at a certain positive time.
To find when the ball hits the ground, we set \( h(t) = 0 \):
\[
160t - 16t^2 = 0
\]
Factoring out \( t \):
\[
t(160 - 16t) = 0
\]
This gives us two solutions:
1. \( t = 0 \) (the starting time)
2. \( 160 - 16t = 0 \) → \( 16t = 160 \) → \( t = 10 \)
Thus, the ball will be in the air from \( t = 0 \) to \( t = 10 \) seconds. Therefore, the reasonable domain of the function is the interval:
\[
0 \leq t \leq 10
\]
This corresponds to option A: {x: 0 ≤ x ≤ 10}.
1. Time \( t \) cannot be negative since we start measuring time from the moment the ball is thrown (which is at \( t = 0 \)).
2. We also need to consider when the ball hits the ground (i.e., when \( h = 0 \)), which occurs at a certain positive time.
To find when the ball hits the ground, we set \( h(t) = 0 \):
\[
160t - 16t^2 = 0
\]
Factoring out \( t \):
\[
t(160 - 16t) = 0
\]
This gives us two solutions:
1. \( t = 0 \) (the starting time)
2. \( 160 - 16t = 0 \) → \( 16t = 160 \) → \( t = 10 \)
Thus, the ball will be in the air from \( t = 0 \) to \( t = 10 \) seconds. Therefore, the reasonable domain of the function is the interval:
\[
0 \leq t \leq 10
\]
This corresponds to option A: {x: 0 ≤ x ≤ 10}.