Problem 83 - College Algebra

Question

On your next vacation, you will divide lodging between large resorts and small inns. Let \(x\) represent the number of nights spent in large resorts. Let \(y\) represent the number of nights spent in small inns. a. Write a system of inequalities that models the following conditions: You want to stay at least 5 nights. At least one night should be spent at a large resort. Large resorts average \(\$ 200\) per night and small inns average \(\$ 100\) per night. Your budget permits no more than \(\$ 700\) for lodging. b. Graph the solution set of the system of inequalities in part (a). c. Based on your graph in part (b), what is the greatest number of nights you could spend at a large resort and still stay within your budget?

Step-by-Step Solution

Answer

By analyzing the modelled scenario through a graph, it's possible to find the maximum number of nights that can be spent at a large resort while still maintaining the budget which is the required solution.

Step 1: Translate scenario into inequalities

By carefully reading the scenario, it can be translated into the following inequalities: \(x + y \geq 5\), which represents staying for at least 5 nights; \(x \geq 1\), which indicates that at least one night should be spent in a large resort; and \(200x + 100y \leq 700\), which represents the budget restriction for lodging.

Step 2: Graph the inequalities

Using a graph and a coordinate system where \(x\) represents large resort stays and \(y\) represents small inn stays, plot the inequalities. Plot each inequality as a line, and then shade the region that fulfills the inequality. The final solution must be the overlap of all individual solutions for each inequality. Note that since \(x\) and \(y\) represent number of nights, they cannot be negative.

Step 3: Interpret the graph to find maximum stays at large resort

With the region representing all solution sets established on the graph, look for the maximum \(x\) value in that region while ensuring that it still falls within the budget constraint. This value of \(x\) is the maximum number of nights that can be spent at a large resort.