Question

student: class: date rational expressions and functions student activity sheet 3; exploring “modeling with rational functions” page 9 of 9 13. reinforce james wants to rent a beach house for the weekend but he cannot afford it alone. he decides to invite some friends to help share the costs. the beach house requires a $2000 deposit, and $450 per guest for the weekend. james and his friends plan to split the total cost evenly. a. write a function c(x) that models the total cost of the beach house in terms of x, the total number of people going. b. write a new function p(x) that models the cost per person for the beach house trip. c. compare the domain of the function model p(x) with the domain of the problem situation.

Explanation:

Step1: Find total cost function

The deposit is a fixed cost of \(2000\). The cost per guest is \(450\), and if there are \(x\) people, the variable cost is \(450x\). So the total cost function \(c(x)=450x + 2000\).

Step2: Find cost - per - person function

The cost per person \(p(x)\) is the total cost divided by the number of people. So \(p(x)=\frac{c(x)}{x}=\frac{450x + 2000}{x}=450+\frac{2000}{x}\).

Step3: Analyze domains

For the function \(p(x)=\frac{450x + 2000}{x}\), the mathematical domain is all real numbers \(x
eq0\) (because division by zero is undefined). In the problem - situation, \(x\) represents the number of people. Since the number of people \(x\) must be a positive integer (\(x\in\mathbb{Z}^+\)), the domain of the problem - situation is a subset of the mathematical domain of \(p(x)\).

Answer:

a. \(c(x)=450x + 2000\)
b. \(p(x)=450+\frac{2000}{x}\)
c. The mathematical domain of \(p(x)\) is \(x
eq0\) (all real numbers except \(0\)). The domain of the problem - situation is \(x\in\mathbb{Z}^+\) (positive integers). The domain of the problem - situation is a subset of the mathematical domain of \(p(x)\)