Problem 55 - Elementary Algebra

Question

A certain cellular phone plan charges \(\$ 23.00\) per month plus \(\$ 0.09\) for each minute of usage. The monthly charge is given by the formula monthly charge \(=0.09 x+23,\) where \(x\) represents the number of minutes of usage per month. What is the charge for a month with 5 hours of usage?

Step-by-Step Solution

Answer

The charge for a month with 5 hours of usage is $50.00.

Step 1: Convert Hours to Minutes

Since the problem states that the usage is given in hours and the formula requires minutes, we need to convert hours to minutes. Given that there are 60 minutes in an hour, 5 hours of usage is equivalent to: \[ 5 \text{ hours} \times 60 \text{ minutes/hour} = 300 \text{ minutes} \]

Step 2: Substitute into the Charge Formula

Now, we substitute the value we found for \(x\) into the monthly charge formula. The formula is: \[ \text{monthly charge} = 0.09x + 23 \] Plugging in \(x = 300\) minutes, the equation becomes: \[ \text{monthly charge} = 0.09 \times 300 + 23 \]

Step 3: Calculate the Usage Charge

Calculate the charge that arises from the minutes of usage by multiplying 0.09 by 300. \[ 0.09 \times 300 = 27 \] This is the charge in dollars for the 300 minutes of usage.

Step 4: Add Base Charge

Now add the base monthly charge of \\(23.00 to the usage charge we calculated: \[ 27 + 23 = 50 \] Hence, the total monthly charge is \\)50.00.

Key Concepts

Rate Calculation

When discussing rate calculation, one is looking to find the cost of something based on a defined rate, such as dollars per minute of phone usage. In this scenario, the rate is \(0.09 per minute. This means that for every minute someone uses their phone, they will be charged \)0.09. It is essential to use the correct rate in all calculations to ensure accurate results.

If the number of minutes changes, the charge will change proportionally. Thus, to calculate the rate charge, multiply the number of minutes by the minute rate of $0.09. This is expressed as:

• If the usage is for 300 minutes, the calculation is: \( 0.09 \times 300 = 27 \text{ dollars} \).

Understanding rate calculations can greatly assist in managing expenses related to time-based services like cellular plans or utility bills.

Unit Conversion

In many mathematical problems, it's necessary to convert units to ensure consistency in calculations. Here, we needed to convert hours into minutes before calculating costs. This process is crucial when working with formulas that require specific units.

To convert from hours to minutes, remember that:

• 1 hour is equivalent to 60 minutes.
• So, 5 hours is converted by multiplying: \( 5 \times 60 = 300 \text{ minutes} \).

This simple but essential step ensures the usage input for any formula that requires minutes is accurate. Neglecting unit conversion can lead to incorrect results, which could affect anything from financial estimates to project timelines.

Problem-Solving Process

Problem-solving in mathematics often involves a systematic series of steps to arrive at a solution. In this exercise, the problem-solving process included identifying the need for unit conversion, rate application, and summation for the total cost.

Here's a breakdown of the problem-solving process used:

• Identify the units that need conversion for the formula — from hours to minutes.
• Substitute the converted value into the given formula — replacing \(x\) with 300.
• Calculate the usage charge — multiplying minute rate by number of minutes.
• Add any fixed charges to the calculated usage charge.

This logical sequence ensures clarity and precision, avoiding errors and ensuring the problem is tackled correctly. Breaking down complex problems into manageable steps is a key skill in solving any mathematical or real-world challenge.