A landscaper mows lawns for at least 3 hours but not more than 6 hours. The landscaper can mow [tex]44,000 \, \text{ft}^2[/tex] per hour. The function [tex]f(t) = 44,000t[/tex] represents the number of square feet the landscaper can mow in [tex]t[/tex] hours.

What is the practical range of the function?

A. All real numbers from 3 to 6, inclusive

B. All multiples of 44,000 between 132,000 and 264,000, inclusive

C. All real numbers

D. All real numbers from 132,000 to 264,000, inclusive

We are given the function

[tex]$$
f(t)=44000t,
$$[/tex]

with the time variable [tex]$t$[/tex] (in hours) restricted to the interval

[tex]$$
3 \leq t \leq 6.
$$[/tex]

Step 1: Identify the Domain

Since the landscaper works for at least 3 hours and at most 6 hours, the domain is

[tex]$$
[3, 6].
$$[/tex]

Step 2: Find the Minimum Value

To find the minimum number of square feet mowed, we evaluate [tex]$f(t)$[/tex] at the smallest value of the domain, [tex]$t=3$[/tex]:

[tex]$$
f(3)=44000 \times 3=132000.
$$[/tex]

Step 3: Find the Maximum Value

To find the maximum number of square feet mowed, we evaluate [tex]$f(t)$[/tex] at the largest value of the domain, [tex]$t=6$[/tex]:

[tex]$$
f(6)=44000 \times 6=264000.
$$[/tex]

Step 4: Determine the Range

Since [tex]$t$[/tex] can be any real number in the interval [tex]$[3,6]$[/tex], the function [tex]$f(t)$[/tex] can produce any value between [tex]$132000$[/tex] and [tex]$264000$[/tex]. Therefore, the range of the function is:

[tex]$$
[132000, 264000].
$$[/tex]

Conclusion

The practical range of the function is "all real numbers from 132,000 to 264,000, inclusive."