A car rental company has $540,000 to purchase up to 25 new cars of two different models. One model costs $18,000 each and the other model costs $24,000 each. Write a system of linear inequalities to describe the situation. Let x represent the first model and y represent the second. Find the region described by the system of linear inequalities.
Well, you know that you have 540,000 dollars to spend, so model x*18,000 + y*24,000 < 540,000.
Next, you know that at most, you may buy 25 new cars, so x + y < 25.
If you treat these two inequalities as equations, you can solve for the extreme points. That is say:
x+y = 25 so
y = 25 - x. Plug y into the first equation and you get 540,000 = x*18,000 + (25-x)*24,000.
Solve for x:
540 = 18x + 600 - 24x
-60 = -6x
x = 10
plug in x=10 into x+y = 25 and you get:
10 + y = 25
y = 15
So, if you buy 10 x model, and 15 y model cars, you'll spend all your money, and have the most cars possible. However, remember that these were inequalities. There is nothing stopping you from spending all your money on model x, or all of it on model y. Also, nothing says that you have to spend all your money either. If you spent all your money on model x, you would get 540,000 / 18,000 = 30 of those. If you spent it all on model y, you would get 22.5 of them. To show the region described by the system, I would make a two sided graph that looks like this:
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|_________| The left vertical line is model x, the right is model y. Going across the bottom is increasing model x and decreasing model y. The y-intercept on the left axis is 30, and the y-intercept on the right axis is 22.5. Where the two cross in the middle is where you get the most car for your money.
Best of luck!
Sorry, the formatting on the graph got messed up. It should be a horizontal line with two vertical lines sticking up at each end.
To write a system of linear inequalities for this situation, we need to consider the constraints given in the problem.
Let x represent the number of cars of the first model and y represent the number of cars of the second model.
The cost of the first model is $18,000 each, so the total cost for x cars of the first model is 18,000x.
The cost of the second model is $24,000 each, so the total cost for y cars of the second model is 24,000y.
According to the problem, the car rental company has $540,000 to purchase up to 25 new cars, so we have the following constraints:
1. The total cost of the first model cars plus the total cost of the second model cars cannot exceed $540,000:
18,000x + 24,000y ≤ 540,000
2. The total number of cars cannot exceed 25:
x + y ≤ 25
These two inequalities represent the system of linear inequalities for the given situation.
To determine the region described by this system of inequalities, we plot the graphs of the equations:
First, let's find the equations of the boundary lines:
1. For the first inequality: 18,000x + 24,000y = 540,000
- Rewrite it in slope-intercept form:
24,000y = -18,000x + 540,000
y = (-18,000/24,000)x + 540,000/24,000
y = -0.75x + 22.5
2. For the second inequality: x + y = 25
- Rewrite it in slope-intercept form:
y = -x + 25
Now, we can graph these two boundary lines on a coordinate plane. The shaded region where the two lines intersect or fall within the inequalities represents the feasible region described by the system.
Here is an example graph depicting the feasible region:
(Horizontal axis = x, Vertical axis = y)
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25| . (7.5,17.5)
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| .
| .
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0 7.5 15 25
In this graph, the shaded area represents the feasible region described by the system of linear inequalities.
Therefore, any combination of x and y values within the shaded region satisfies the given constraints and represents a valid solution for the car rental company.