Brenda likes Coca-Cola and Hot dogs. Coca-Cola cost $4 per bottle and Hot dogs cost $2 each. There is a special promotion for Coke that will last for one month. If Brenda sends in the bottle tops from the Cokes she drinks during the next month, she will get a refund of $2 for every bottle cap beyond the first 40 that she returns. For example, if she returns 45 bottle caps she will get back $10 = 2(45 − 40). Barbara has $200 to spend on Coke and Hot dogs during the next month, plus the money she gets back from the bottle tops.
(a) Draw her budget line with Coke on the horizontal axis and hot dogs on the vertical axis.
(b) Find the points where the budget line hits the axes and the point where it has a kink. At each of these three points write down the quantities of each good consumed.
(a) The budget line can be drawn by plotting different combinations of Coca-Cola and Hot dogs that Brenda can purchase with her given budget of $200.
Let's assume the quantity of Coca-Cola (in bottles) is denoted by x, and the quantity of Hot dogs is denoted by y.
Given that Coca-Cola costs $4 per bottle and Hot dogs cost $2 each, the budget line can be represented by the equation:
4x + 2y = 200
(b) To find the points where the budget line hits the axes, we can set one of the variables to 0 and solve for the other variable.
When x = 0, we have 2y = 200, which gives y = 100. So, one point is (0, 100).
Similarly, when y = 0, we have 4x = 200, which gives x = 50. So, another point is (50, 0).
The point where the budget line has a kink occurs when Brenda returns her bottle tops and gets the refund.
According to the given information, Brenda gets a refund of $2 for every bottle cap beyond the first 40 that she returns. This means she will get back $2 for each additional bottle cap, resulting in a linear increase in her budget.
Let's assume Brenda returns n bottle tops. Her total refund amount will be calculated as 2 * (n - 40). Adding this refund to her initial budget of $200, we get her new budget:
New budget = 200 + 2 * (n - 40)
To find the point where the budget line has a kink, we need to find the value of n that results in a new budget where she can only buy Coca-Cola or Hot dogs (not both). This means one of the quantities (x or y) should be 0.
Considering only Coca-Cola, we can set y = 0 in our budget line equation:
4x + 2(0) = 200 + 2 * (n - 40)
4x = 200 + 2 * (n - 40)
4x = 200 + 2n - 80
4x = 2n + 120
Simplifying this equation, we get: x = 0.5n + 30
Now, we can substitute this equation into our budget line equation to find the n value:
4(0.5n + 30) + 2y = 200
2n + 240 + 2y = 200
2n + 2y = -40
n + y = -20
From this equation, we can see that there is no positive value of n that satisfies it. Therefore, there is no kink point on the budget line.