Final answer:
By setting up linear equations for each limousine company based on their hourly rates and fixed fees, we can solve for the number of hours where the cost is the same for both. Ms. Chen needs to rent the limousine for 4 hours for the costs to be equal.
Step-by-step explanation:
To determine the number of hours Ms. Chen needs to rent the limousine for the cost to be the same at both Company A and Company B, we can set up two linear equations based on the given information. Let x represent the number of hours the limousine is rented, and let y represent the total cost.
For Company A, the cost for 2 hours is $550 and for 6 hours it's $1,550. This information gives us two points: (2, 550) and (6, 1550). We can use these to find the equation of the line in the form of y = mx + b, where m is the hourly rate and b is the set fee. By calculating the difference in cost and dividing by the difference in hours, we find that Company A charges $250 per hour. The set fee can be found by substituting one of the points into the equation and solving for b, which gives us a set fee of $50.
Similarly, for Company B, the cost for 2 hours is $600 and the cost for 6 hours is $1,500. Following the same process, Company B's hourly rate is $225, and the set fee is $150.
Now, we want to find the number of hours where the cost from both companies is equal, so we set the two equations equal to each other:
Company A: y = 250x + 50
Company B: y = 225x + 150
Setting them equal gives us 250x + 50 = 225x + 150. Solving for x:
25x = 100
x = 4
Ms. Chen needs to rent the limousine for 4 hours for the costs from both companies to be the same.