Problem 20 - Precalculus Mathematics for Calculus

Question

A cell phone company charges a monthly fee of \(\$ 10\) for the first 1000 text messages and 10 cents for each additional text message. Miriam's bill for text messages for the month of June is \(\$ 38.50 .\) How many text messages did she send that month?

Step-by-Step Solution

Answer

Miriam sent 1285 text messages.

Step 1: Identify the Total Cost Components

Miriam's total bill is $38.50. This includes a base fee of $10 for the first 1000 text messages and an additional charge for any text messages beyond 1000.

Step 2: Calculate the Excess Charge

Subtract the base fee from Miriam's total bill to find the amount charged for additional text messages. \[ 38.50 - 10 = 28.50 \] So, $28.50 is the charge for the extra messages.

Step 3: Determine the Number of Additional Texts

The charge per additional text message is 10 cents. Convert $28.50 into cents to ease the calculation:\[ 28.50 \text{ dollars} \times 100 = 2850 \text{ cents} \]Now, divide the total additional cost by the cost per additional text message:\[ 2850 \div 10 = 285 \]This means Miriam sent 285 additional text messages beyond the initial 1000.

Step 4: Calculate the Total Number of Texts

Add the 1000 text messages included in the base fee to the 285 additional text messages:\[ 1000 + 285 = 1285 \]Miriam sent a total of 1285 text messages.

Key Concepts

Linear Equations

Linear equations are a fundamental concept in algebra, allowing us to find unknown values by relating them to known quantities. When solving problems, they often appear in simple forms like \(ax + b = c\), where \(a\) and \(b\) are constants, and \(x\) is the variable we're looking to solve. In the context of a real-world scenario, such as calculating cellphone charges, linear equations help us simplify and solve complex billing problems. Here are key points about linear equations:

• Linear equations represent a straight line when graphed, making it easy to visualize the relationship between variables.
• To solve a linear equation, we aim to isolate the variable on one side of the equation, thus revealing its value.
• Each step in solving a linear equation involves performing the same operation on both sides to maintain balance.

Understanding linear equations is crucial for analyzing and efficiently solving problems related to real-world scenarios such as billing analyses, like Miriam's text messaging case.

Word Problems

Word problems in algebra require translating a real-world scenario into a mathematical model before solving it. They often combine words with numbers, posing unique challenges but also providing a practical context. Let's explore what makes word problems special:

• They require careful reading to identify quantities, operations, and relationships described in words.
• The process involves defining variables that represent particular quantities to formulate a mathematical equation.
• Solving word problems improves understanding by applying mathematical concepts to everyday situations.

In the problem with Miriam's cellphone bill, extracting key pieces of information like the base charge and the cost per additional text was necessary to establish the equation to find the total number of texts she sent.

Cost Analysis

Cost analysis in real-world problems involves calculating and understanding different components that contribute to the total cost. This process helps identify where changes could be made to affect overall spending, making it vital for budgeting and decision-making. Here are some insights into cost analysis within this context:

• It typically breaks down into fixed costs, like the base fee for the first 1000 text messages, and variable costs, such as charges for extra texts.
• Understanding how each cost component affects the total can help make informed decisions about usage and savings.
• Detailed cost analysis exposes potential for optimization, showing us, for example, how managing text use could lower monthly bills.

In Miriam's situation, understanding the breakdown of her phone bill into base and additional charges allowed for precise calculation of her total text messages in June. This demonstrates how cost analysis extends beyond solving equations and into practical budgeting and financial planning.