Problem 54 - Algebra 2

Question

AUDIO BOOK DOWNLOADS Chaz has a collection of 15 audio books. After he gets a part-time job, he decides to download 3 more audio books each month. The function \(A(t)=15+3 t\) counts the number of audio books \(A(t)\) he has after \(t\) months. How many audio books will he have after 8 months?

Step-by-Step Solution

Answer

Chaz will have 39 audio books after 8 months.

Step 1: Understand the Function

The function given is \(A(t) = 15 + 3t\). This tells us that the initial number of audio books is 15, and every month \(t\) he downloads 3 more.

Step 2: Substitute the Variable t

We need to find the number of audio books after 8 months. Substitute \(t = 8\) into the function: \(A(8) = 15 + 3 \, \times \, 8\).

Step 3: Calculate the Expression

Calculate the expression by evaluating the multiplication and addition: \(A(8) = 15 + 24\).

Step 4: Perform the Addition

Add the numbers to find the total number of audio books: \(A(8) = 39\).

Key Concepts

Variable Substitution

Variable substitution is a key concept in working with functions and equations. In the context of this exercise, it allows us to determine the number of audio books Chaz will have in the future by replacing a variable in the function with a specific value. This process involves

• Identifying the variable that represents the changing quantity, which in this case is time in months, denoted by \( t \).
• Selecting a specific value for the variable you want to examine, here \( t = 8 \), to determine the scenario after 8 months.
• Substituting this value into the function \( A(t) = 15 + 3t \) to find out \( A(8) \).

This substitution process transforms the general function into a specific scenario, giving us the ability to make predictions based on real-world conditions.

Function Evaluation

Function evaluation is the process of calculating an expression by substituting inputs into a function. For Chaz's audio book situation, after substituting \( t = 8 \) into the function \( A(t) = 15 + 3t \), we evaluate the expression. Here's how we do it:

• First, substitute 8 into the function: \( A(8) = 15 + 3 \times 8 \).
• Next, carry out the multiplication: \( 3 \times 8 = 24 \).
• Add the result to the initial 15 audio books: \( 15 + 24 = 39 \).

This evaluation shows that Chaz will have 39 audio books after 8 months. Function evaluation is crucial as it helps in understanding the implications of a model, allowing for practical insights from mathematical expressions.

Mathematical Modeling

Mathematical modeling involves creating representations of real-world situations using mathematical language and structures. The equation \( A(t) = 15 + 3t \) is a model for Chaz’s growing collection of audio books. It helps us:

• Understand how a situation evolves over time through a simple linear function.
• Predict future outcomes, like finding the number of audio books after a certain period, such as 8 months.
• Make informed decisions based on the model, like planning when to stop downloading more books.

The linear function accurately captures the constant rate of change (3 books per month), making mathematical modeling an effective tool for simplifying and solving real-life problems through mathematical analysis.