The cost of a telephone call between two cities is $0.30 for the first minute and $0.16 for each additional minute or fraction thereof.

Question

The cost of a telephone call between two cities is $0.30 for the first minute and $0.16 for each additional minute or fraction thereof. A formula for the cost is given below, where t is the time in minutes. (Note: [[x]] = greatest integer n such that n ≤ x. For example [[3.2]] = 3 and [[-1.6]] = -2.)

C(t) = 0.30 – 0.16[[ – (t – 1)]]

(a) Use a graphing utility to graph the cost function for 0 < t ≤ 5.

(b) Use the graph to complete the table below and observe the behavior of the function as t approaches 3.5.

t	3	3.3	3.4	3.5	3.6	3.7	4
C(t)

Use the graph and the table to find the following limit.

(c) Use the graph to complete the table below and observe the behavior of the function as t approaches 3.

t	2	2.5	2.9	3	3.1	3.5	4
C(t)

Does the limit of C(t) as t approaches 3 exist?

The limit does not exist.The limit exists.    

Answer 1

Hi Kaitlyn,

Looking at the four provided graphs in subsection (a) - the top two graphs can immediately be excluded as incorrect because the curve needs to start at 0.30. To start the phone call costs $0.30 - so the graphs that start at 1 must be incorrect.

The difference between the two lower graphs is related to the open or closed endpoints. As is visible on the graphs the circle that left (lower) side of each segment is a closed (filled in) red dot on the graph to the right and an open dot on the graph to the left. This signifies that the graph on the right includes the y-value of the segment at the end x-value of the segment. In other words, the graph on the left would have a value of y = 0.3 AND 0.46 at x = 1. The graph on the right would only have a value of y = 0.3 because the segment that spans x = 1 to x = 2 and has a y-value of 0.46 has an open circle at x = 1. Looking back at the problem wording, this is consistent with the problem statement. For a call of exactly 1 minute, the cost would be 30 cents. For a call of exactly 2 minutes, the cost would be 46 cents, etc.

For section (b) - the cost of a call of exactly 3 minutes would be 0.62, but going to the next call duration of 3.3 would cost 0.78. This will be true for all other values of the table, including call duration of 4 minutes. The limit as t approaches 3.5 would then be 0.78. From either direction that you approach 3.5 all the values in the table are 0.78.

For section (c) - 2 minutes is 46 cents, but 2.5 minutes is bumped up to 0.62, and this value would be consistent right up to 3 minutes, the same as in section (b). For the cost of the call from 3.1 to 4 the call is 0.78. So the values in your table change part way through. If you approach t = 3 from the left (smaller numbers) - then all values are 0.46 but if you approach from the right (bigger numbers) the values are all 0.78. This would mean that there is no single limit as "t" approaches 3.