Problem 109
Question
Telephone Charges A long distance phone service charges \(\$ 0.40\) for the first 10 minutes and \(\$ 0.05\) for each additional minute or fraction thereof. Use the greatest integer function to write the cost \(C\) of a call in terms of time \(t\) (in minutes). Sketch the graph of this function and discuss its continuity.
Step-by-Step Solution
Verified Answer
The cost function is \(C(t) = 0.40 + 0.05(\lceil t \rceil - 10)\) for \(t > 10\) minutes, and \(C(t) = 0.40\) for \(t \leq 10\) minutes. The graph of the function is a step graph, indicating a rapid increase in the cost after 10 minutes and is discontinuous at every step.
1Step 1: Understand and formulate the cost function
To find the cost function, first understand the different charges. There is a flat rate of \$0.40 for the first 10 minutes, then an additional \$0.05 for each additional minute. This can be written as: \(C(t) = 0.40 + 0.05(\lceil t \rceil - 10)\) when \(t > 10\) minutes, and \(C(t) = 0.40\) when \(t \leq 10\) minutes.
2Step 2: Sketch the graph and discuss its continuity
To sketch the graph y = C(t), plot points for the initial 10 minutes at \$0.40, and then plot the increments of \$0.05 for every additional minute. This results in a step graph, which rapidly increases in steps after t = 10 minutes. This type of function is discontinuous at every step, but is continuous within each step.
Key Concepts
Cost Function
Cost Function
When we talk about a cost function in the context of telephone charges, we're referring to a mathematical way of representing how the total cost of the call increases with its duration. Starting with the scenario given, we identify two pricing stages: a flat rate for the initial period and a variable rate afterwards.
In simple terms, if the call lasts for 10 minutes or less, the cost is a fixed amount, which we call a base charge. Beyond that, every additional minute—no matter how small—involves a small incremental charge. The cost function for this scenario combines these two stages and is expressed as follows: for a call duration of time t (in minutes), the cost C is
In simple terms, if the call lasts for 10 minutes or less, the cost is a fixed amount, which we call a base charge. Beyond that, every additional minute—no matter how small—involves a small incremental charge. The cost function for this scenario combines these two stages and is expressed as follows: for a call duration of time t (in minutes), the cost C is
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