Solve each problem. The cost to hire a caterer for a party...

Solve each problem. The cost to hire a caterer for a party depends on the number of guests attending. If 100 people attend, the cost per person will be 20 dollars. For each person less than 100 dollars, the cost will increase by 5 . Assume that no more than 100 people will attend. Let $x$ represent the number less than 100 who do not attend. For example, if 95 attend, then $x=5$ (a) Write a function $N(x)$ giving the number of guests. (b) Write a function $G(x)$ giving the cost per guest. (c) Write a function $N(x) \cdot G(x)$ for the total cost, $C(x)$ (d) What is the total cost if 80 people attend?

Solve each problem.

The cost to hire a caterer for a party depends on the number of guests attending. If 100 people attend, the cost per person will be 20 dollars. For each person less than 100 dollars, the cost will increase by 5 . Assume that no more than 100 people will attend. Let $x$ represent the number less than 100 who do not attend. For example, if 95 attend, then $x=5$
(a) Write a function $N(x)$ giving the number of guests.
(b) Write a function $G(x)$ giving the cost per guest.
(c) Write a function $N(x) \cdot G(x)$ for the total cost, $C(x)$
(d) What is the total cost if 80 people attend?

Key Concepts

Linear Functions

This concept involves creating functions that describe relationships where the rate of change is constant. In the context of cost modeling, a linear function can be used to represent how the cost per guest increases or decreases as the number of guests varies. The slope of the function represents the rate at which the cost changes, and the y-intercept represents the initial cost under standard conditions.

Variable Identification and Domain Modeling

Here, the variable is defined based on the deviation from a standard quantity, and understanding its meaning is crucial to setting up correct functions. Identifying what the variable represents aids in accurately determining the function's domain and ensuring appropriate application in real-world conditions, such as limiting attendance or cost calculations.

Composite Functions and Function Multiplication

This involves combining two or more functions to form a new function. In modeling a real-world scenario, the overall cost may be represented as a product of a function that calculates the number of guests and another that calculates the cost per guest. This multiplication leads to a composite function that provides an overall measure, such as the total cost.

Translating Word Problems into Algebraic Models

This key concept is about interpreting the details and constraints provided in a word problem and converting them into mathematical expressions or functions. It requires an understanding of the relationships between different quantities described in the problem, which is essential in accurately modeling real-life scenarios such as pricing or budgeting.

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