5. DETAILS TAMUBUSMATH 5.5.008. 0/6 Submissions Used A restaurant offers a catering service which costs $\$ 18.50$ per person with a $\$ 74.75$ service charge. For parties of 50 or more people, a group discount applies, and the cost is $\$ 13.75$ per person along with the service charge dropping to $\$ 37.00$. Write a piecewise-defined function which calculates the total cost, $T$, (in dollars) of the catering service, which serves $n$ people. \[ T(n)=\left\{\begin{array}{ll} \square & \text { in } \square, \square=\square \\ \square & \text { in } n \geq \square \end{array}\right. \] Submit Answer MY NOTES
Solution
Step 1 ：The problem is asking for a piecewise function that calculates the total cost of the catering service based on the number of people served. The function will have two parts: one for parties of less than 50 people and one for parties of 50 or more people.
Step 2 ：For parties of less than 50 people, the cost is $18.50 per person plus a $74.75 service charge. So, the function for this case would be \(T(n) = 18.50n + 74.75\).
Step 3 ：For parties of 50 or more people, the cost is $13.75 per person and the service charge is $37.00. So, the function for this case would be \(T(n) = 13.75n + 37.00\).
Step 4 ：So, the piecewise function would be: \[T(n)=\left\{\begin{array}{ll} 18.50n + 74.75 & \text { for } n < 50 \\ 13.75n + 37.00 & \text { for } n \geq 50 \end{array}\right.\]
Step 5 ：Final Answer: \[\boxed{T(n)=\left\{\begin{array}{ll} 18.50n + 74.75 & \text { for } n < 50 \\ 13.75n + 37.00 & \text { for } n \geq 50 \end{array}\right.}\]
