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
what is the domain
what is the domain
Answers
GPT-4o mini
To determine the domain of the function \( H(t) = 400 - 16t^2 \), we need to consider the physical situation described by the model.
1. **Physical Context:** The function represents the height of a ball dropped from a height of 400 feet. The time \( t \) starts at 0 seconds when the ball is dropped.
2. **When does the ball hit the ground?** The ball will hit the ground when \( H(t) = 0 \). We can find the time at which this occurs by solving the equation:
\[
400 - 16t^2 = 0
\]
Rearranging gives us:
\[
16t^2 = 400
\]
\[
t^2 = 25
\]
\[
t = 5 \quad (\text{since time cannot be negative})
\]
3. **Domain:** The ball is in free fall from \( t = 0 \) seconds until it hits the ground at \( t = 5 \) seconds. Therefore, the time \( t \) must satisfy:
\[
0 \leq t \leq 5
\]
In interval notation, the domain of the function \( H(t) \) is:
\[
[0, 5]
\]
Thus, the domain of the function is \( [0, 5] \).
1. **Physical Context:** The function represents the height of a ball dropped from a height of 400 feet. The time \( t \) starts at 0 seconds when the ball is dropped.
2. **When does the ball hit the ground?** The ball will hit the ground when \( H(t) = 0 \). We can find the time at which this occurs by solving the equation:
\[
400 - 16t^2 = 0
\]
Rearranging gives us:
\[
16t^2 = 400
\]
\[
t^2 = 25
\]
\[
t = 5 \quad (\text{since time cannot be negative})
\]
3. **Domain:** The ball is in free fall from \( t = 0 \) seconds until it hits the ground at \( t = 5 \) seconds. Therefore, the time \( t \) must satisfy:
\[
0 \leq t \leq 5
\]
In interval notation, the domain of the function \( H(t) \) is:
\[
[0, 5]
\]
Thus, the domain of the function is \( [0, 5] \).