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Answer :
To determine the practical domain of the function [tex]\( f(x) = -85x + 3145 \)[/tex], we need to consider what [tex]\( x \)[/tex] represents in the context of the problem. Here, [tex]\( x \)[/tex] stands for the number of minutes after printing begins.
The function models the number of unprinted pages. The goal is to understand during which time intervals this function makes sense for the given problem. Let's break it down:
1. Initial Condition: At [tex]\( x = 0 \)[/tex], no time has passed, so all 3145 pages are unprinted. This means [tex]\( f(0) = 3145 \)[/tex].
2. Final Condition: We want to know how long it takes to print all 3145 pages, which is the point where there are no unprinted pages left, i.e., [tex]\( f(x) = 0 \)[/tex].
3. Solve for [tex]\( x \)[/tex]:
[tex]\[
f(x) = -85x + 3145 = 0
\][/tex]
[tex]\[
-85x = -3145
\][/tex]
[tex]\[
x = \frac{3145}{85}
\][/tex]
4. Calculate the Time:
[tex]\[
x = 37
\][/tex]
It takes 37 minutes to print all the pages.
5. Practical Domain: The practical domain includes all the whole number minutes from the start until the task is done. Since printing must proceed in whole minutes, the practical domain is the set of whole numbers from 0 to 37.
Therefore, the practical domain of the function [tex]\( f(x) \)[/tex] is all whole numbers from 0 to 37.
The function models the number of unprinted pages. The goal is to understand during which time intervals this function makes sense for the given problem. Let's break it down:
1. Initial Condition: At [tex]\( x = 0 \)[/tex], no time has passed, so all 3145 pages are unprinted. This means [tex]\( f(0) = 3145 \)[/tex].
2. Final Condition: We want to know how long it takes to print all 3145 pages, which is the point where there are no unprinted pages left, i.e., [tex]\( f(x) = 0 \)[/tex].
3. Solve for [tex]\( x \)[/tex]:
[tex]\[
f(x) = -85x + 3145 = 0
\][/tex]
[tex]\[
-85x = -3145
\][/tex]
[tex]\[
x = \frac{3145}{85}
\][/tex]
4. Calculate the Time:
[tex]\[
x = 37
\][/tex]
It takes 37 minutes to print all the pages.
5. Practical Domain: The practical domain includes all the whole number minutes from the start until the task is done. Since printing must proceed in whole minutes, the practical domain is the set of whole numbers from 0 to 37.
Therefore, the practical domain of the function [tex]\( f(x) \)[/tex] is all whole numbers from 0 to 37.
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