Answer :
To solve the problem of determining which function [tex]\( f \)[/tex] represents the total charge for a tour group with [tex]\( n \)[/tex] people, where [tex]\( n \geq 25 \)[/tex], let's break it down step-by-step.
1. Determine the cost for the first 25 people:
- The museum charges \[tex]$21 per person for the first 25 people. So, the cost for these 25 people is:
\[ 25 \times 21 = 525 \text{ dollars} \]
2. Determine the cost for additional people beyond the first 25:
- For each additional person (beyond the 25), the cost is \$[/tex]14 per person.
3. Express the total cost as a function of [tex]\( n \)[/tex]:
- If there are [tex]\( n \)[/tex] people, the number of additional people beyond the first 25 is [tex]\( n - 25 \)[/tex].
- The cost for these additional people is [tex]\( 14 \times (n - 25) \)[/tex].
4. Calculate the total cost function:
- The total cost [tex]\( f(n) \)[/tex] for [tex]\( n \)[/tex] people (where [tex]\( n \geq 25 \)[/tex]) is the sum of the cost for the first 25 people and the additional people.
- So the total function becomes:
[tex]\[ f(n) = 525 + 14 \times (n - 25) \][/tex]
5. Simplify the function:
- Simplifying [tex]\( f(n) = 525 + 14 \times (n - 25) \)[/tex]:
[tex]\[
f(n) = 525 + 14n - 350 = 14n + 175
\][/tex]
Therefore, the function [tex]\( f(n) = 14n + 175 \)[/tex] describes the total charge, in dollars, for a tour group with [tex]\( n \)[/tex] people, where [tex]\( n \geq 25 \)[/tex].
The correct answer is option A: [tex]\( f(n) = 14n + 175 \)[/tex].
1. Determine the cost for the first 25 people:
- The museum charges \[tex]$21 per person for the first 25 people. So, the cost for these 25 people is:
\[ 25 \times 21 = 525 \text{ dollars} \]
2. Determine the cost for additional people beyond the first 25:
- For each additional person (beyond the 25), the cost is \$[/tex]14 per person.
3. Express the total cost as a function of [tex]\( n \)[/tex]:
- If there are [tex]\( n \)[/tex] people, the number of additional people beyond the first 25 is [tex]\( n - 25 \)[/tex].
- The cost for these additional people is [tex]\( 14 \times (n - 25) \)[/tex].
4. Calculate the total cost function:
- The total cost [tex]\( f(n) \)[/tex] for [tex]\( n \)[/tex] people (where [tex]\( n \geq 25 \)[/tex]) is the sum of the cost for the first 25 people and the additional people.
- So the total function becomes:
[tex]\[ f(n) = 525 + 14 \times (n - 25) \][/tex]
5. Simplify the function:
- Simplifying [tex]\( f(n) = 525 + 14 \times (n - 25) \)[/tex]:
[tex]\[
f(n) = 525 + 14n - 350 = 14n + 175
\][/tex]
Therefore, the function [tex]\( f(n) = 14n + 175 \)[/tex] describes the total charge, in dollars, for a tour group with [tex]\( n \)[/tex] people, where [tex]\( n \geq 25 \)[/tex].
The correct answer is option A: [tex]\( f(n) = 14n + 175 \)[/tex].
