Question

A student is planning a trip to Dubai and wants to budget their expenses. They have a maximum of AED 500 for accommodation and AED 300 for food. The student wants to stay in a hotel that charges at most 200 AED per night and eat at a restaurant that charge at most AED 50 per meal. Write a system of linear inequalities to represent this situation and find the feasible region

Answer 1

Bard here! Your answer is....

The system of linear inequalities that represents the student's budget constraints is:

x≤2.5

y≤6

x≥0

y≥0

Explanation:

Let x represent the number of nights the student stays at the hotel and y represent the number of meals they eat at the restaurant.

Accommodation:

The student has a maximum budget of AED 500 for accommodation, and the hotel charges at most AED 200 per night. Therefore, the total cost of accommodation should not exceed AED 500:

200x≤500

Dividing both sides by 200, we get:

x≤2.5

Food:

The student has a maximum budget of AED 300 for food, and the restaurant charges at most AED 50 per meal. Therefore, the total cost of food should not exceed AED 300:

50y≤300

Dividing both sides by 50, we get:

y≤6

Non-negativity:

The number of nights and meals cannot be negative:

x≥0

y≥0

Feasible Region:

The feasible region is the area where all the inequalities are satisfied. It is represented by the shaded area in the graph below.

graph with xaxis representing the number of nights and yaxis representing the number of meals. The shaded area is bounded by the lines x = 2.5, y = 6, x = 0, and y = 0.

Graph with xaxis representing the number of nights and yaxis representing the number of meals. The shaded area is bounded by the lines x = 2.5, y = 6, x = 0, and y = 0.

The feasible region includes all points within the shaded area, including the boundary lines. It represents all possible combinations of nights and meals that the student can afford within their budget.

Bard is always happy to help!

Answer 2

Let's define some variables to represent the unknowns in this problem:

Let:

x = number of nights spent in the hotel

y = number of meals eaten at restaurants

Now, let's set up the system of linear inequalities based on the given information:

1. The maximum amount for accommodation is AED 500, and the hotel charges at most AED 200 per night:

200x ≤ 500

2. The maximum amount for food is AED 300, and the restaurant charges at most AED 50 per meal:

50y ≤ 300

3. Additionally, we have the constraints that x and y should be non-negative:

x ≥ 0

y ≥ 0

Now, let's plot these inequalities on a graph to find the feasible region:

The first inequality, 200x ≤ 500, can be rewritten as x ≤ 2.5.

The second inequality, 50y ≤ 300, can be rewritten as y ≤ 6.

We also know that x and y should be non-negative, so we have x ≥ 0 and y ≥ 0.

Plotting these constraints on a graph, we find that the feasible region is a triangular region bounded by the x-axis, y-axis, and the lines x = 0. y = 0, x = 2.5, and y = 6.