Problem 60 - Introductory and Intermediate Algebra for College Students 4th

Question

Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. The rate for a particular international telephone call is \(\$ 0.55\) for the first minute and \(\$ 0.40\) for each additional minute. Determine the length of a call that costs \(\$ 6.95\)

Step-by-Step Solution

Answer

The length of a call that costs \$6.95 is 17 minutes.

Step 1: Calculate remaining amount after first minute

The first minute of the call costs \$0.55. Therefore, after the first minute the remaining balance is \$6.95 - \$0.55 = \$6.40.

Step 2: Determine the number of additional minutes

Each additional minute of the call costs \$0.40. So, to find out how many additional minutes the remaining amount will cover, we perform the following operation: \$6.40 / \$0.40 = 16 minutes.

Step 3: Calculate the total duration of the call

Adding the first minute and the additional minutes, we get the total duration of the call: 1 (first minute) + 16 (additional minutes) = 17 minutes.

Key Concepts

Problem Solving

Problem solving is an essential skill in intermediate algebra. When facing a complex problem, breaking it down into smaller, manageable steps can be very helpful. In this telephone call exercise, we need to determine the total length of a call based on its cost. The challenge here requires you to interpret the given rates and costs.

We start by understanding what the problem is asking: "What is the total call length if the call costs $6.95?" Once this question is clear, we can plan our steps. The idea is to first handle the fixed cost of the first minute and then move on to the variable cost of additional minutes. This structured approach allows us to tackle the problem systematically without skipping any steps.

Rate Calculation

Rate calculation is the principle of determining the cost associated with using or consuming a resource over time. In this exercise, we are given two rates:

• $0.55 for the first minute of the call
• $0.40 for each additional minute.

This information helps us calculate the remaining expenditure after the first minute. Initially, the total cost is $6.95. We subtract the first minute's rate to determine how much money is left for the additional minutes. Understanding rate calculation is crucial because it allows us to evaluate the remaining amount that can be spent at this variable rate, which directly affects how many additional minutes we can afford.

Cost Analysis

Cost analysis involves evaluating all aspects of cost in decision-making. In our exercise, understanding how much each component of the call costs helps us decide how to allocate the $6.95 across the duration of the call.

The cost analysis starts by acknowledging the fixed cost: the $0.55 fee for the first minute of the call. Knowing this upfront allows us to allocate $0.40 for each subsequent minute out of the remaining $6.40. Performing these calculations step-by-step gives us clarity and avoids errors in estimating how much time is covered by the remaining funds.

Equation Solving

Equation solving is a method to find unknowns using mathematical expressions. In this scenario, creating an equation can assist us in solving for the unknown call length. We know:

• Cost for the first minute: \(0.55
• Total cost: \)6.95

We set up the equation by expressing the additional minutes as a variable, which we multiply by the additional minute rate of \(0.40, then add the initial \)0.55. The equation looks like this: \[ 0.55 + 0.40x = 6.95 \] Solving for \(x\), this approach lets us find how many additional minutes can be purchased once the initial minute is paid for. It further reaffirms the power of equation solving in problem-solving contexts, especially when dealing with finances or rates.