A restaurant offers a catering service which costs $20.00 per person with a $106.50 service charge. For parties of 60 or more people, a group discount applies, and the cost is $15.25 per person along with the service charge dropping to $53.00. Write a piecewise-defined linear function which calculates the total cost T of the catering service which serves n people.
T(n)
if
≤n ≤
if n >

To write the piecewise-defined linear function $ T(n) $ for the total cost of the catering service, we need to account for two scenarios:

1. For parties of fewer than 60 people, the cost per person is $20.00 and the service charge is $106.50.
2. For parties of 60 or more people, the cost per person is $15.25 and the service charge drops to $53.00.

Thus, the function $ T(n) $, where $ n $ represents the number of people, can be defined as:

\[
T(n) =
\begin{cases} 
20n + 106.50 & \text{if } 1 \leq n < 60 \\
15.25n + 53.00 & \text{if } n \geq 60 
\end{cases}
\]

### Explanation:
- For $ 1 \leq n < 60 $: The cost is $ 20n + 106.50 $.
- For $ n \geq 60 $: The cost is $ 15.25n + 53.00 $.

Would you like any further clarification or examples to help understand this function better?

### Related Questions:
1. How would the total cost change if the service charge for fewer than 60 people increased?
2. What would happen if the group discount applied to groups of 50 or more people instead of 60?
3. How does the total cost vary when the number of people is exactly 60?
4. What would be the cost for a party of 45 people?
5. How much money does a group of 75 people save compared to a group of 59 people?

### Tip:
Piecewise functions are great for modeling real-world situations where different conditions lead to different outcomes!

A restaurant offers a catering service which costs $20.00 per person with a $106.50 service charge. For parties of 60 or more people, a group discount applies, and the cost is $15.25 per person along with the service charge dropping to $53.00. Write a piecewise-defined linear function which calculates the total cost T of the catering service which serves n people.

Learn how to write a piecewise-defined linear function to calculate the total cost of a catering service. This problem involves algebraic concepts like linear functions and piecewise functions, suitable for students in Grades 9-12. It covers two pricing models depending on the number of people in the party.

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