To solve the problem, we need to determine the practical range of the function [tex]\( f(t) = 2.7t \)[/tex], which represents the distance Hong hikes in [tex]\( t \)[/tex] hours. We are given that Hong hikes at least 1 hour but no more than 4 hours, and her average hiking rate is 2.7 miles per hour.
Let's analyze the function and the given constraints step by step:
1. Understanding the function: The function [tex]\( f(t) = 2.7t \)[/tex] indicates the distance Hong hikes based on the time [tex]\( t \)[/tex] in hours. The rate, 2.7 miles per hour, is multiplied by the time to give the total distance.
2. Identifying the domain: The domain of our function, which represents the possible values for [tex]\( t \)[/tex], is given as [tex]\( 1 \leq t \leq 4 \)[/tex]. This means [tex]\( t \)[/tex] can be any real number between 1 and 4, inclusive.
3. Calculating the minimum distance: To find the minimum distance Hong can hike, we substitute the smallest value of [tex]\( t \)[/tex] into the function:
[tex]\[
f(1) = 2.7 \times 1 = 2.7 \text{ miles}
\][/tex]
4. Calculating the maximum distance: To find the maximum distance Hong can hike, we substitute the largest value of [tex]\( t \)[/tex] into the function:
[tex]\[
f(4) = 2.7 \times 4 = 10.8 \text{ miles}
\][/tex]
5. Determining the practical range: The practical range of the function [tex]\( f(t) \)[/tex] is the set of all possible distances that Hong can hike given the time constraints. Since [tex]\( t \)[/tex] ranges from 1 to 4, the distance [tex]\( f(t) \)[/tex] will range from 2.7 miles to 10.8 miles.
Therefore, the practical range of the function is all real numbers from 2.7 to 10.8, inclusive.
So, the correct answer is:
- All real numbers from 2.7 to 10.8, inclusive
