Alright, let's tackle this problem step-by-step to understand how we derive the correct function notation.
1. Identify the given information:
- You usually maintain a speed of 3 miles per hour.
- [tex]$t$[/tex] is the independent variable, representing time in hours.
- [tex]$h(t)$[/tex] is the dependent variable, representing the distance traveled in miles.
2. Understand the relationship between speed, time, and distance:
- Speed is the rate at which distance changes over time.
- Distance traveled can be calculated with the formula:
[tex]\[
\text{Distance} = \text{Speed} \times \text{Time}
\][/tex]
- In this case, speed is 3 miles per hour.
3. Set up the function:
- We are asked to express the distance traveled ([tex]$h(t)$[/tex]) as a function of time ([tex]$t$[/tex]).
- Using the given speed, we can write the distance as:
[tex]\[
h(t) = 3 \times t
\][/tex]
- Here, [tex]$h(t)$[/tex] represents the distance, and it is equal to the product of the speed (3 miles per hour) and the time ([tex]$t$[/tex] hours).
4. Match the functional relationship to the given options:
- Option a: [tex]\(h(t) = 3h\)[/tex] ⇒ This incorrectly uses [tex]$h$[/tex] on the right-hand side.
- Option b: [tex]\(h(t) = 3t\)[/tex] ⇒ This correctly represents the relationship where [tex]$t$[/tex] is the independent variable.
- Option c: [tex]\(t(h) = 3t\)[/tex] ⇒ This incorrectly places [tex]$t$[/tex] as the dependent variable.
- Option d: [tex]\(t(h) = 3h\)[/tex] ⇒ This incorrectly changes the roles of the variables.
Thus, the correct function notation that describes the distance traveled as a function of time is:
[tex]\[ \boxed{h(t) = 3t} \][/tex]
Therefore, the correct statement is option b.
