Problem 9 - Algebra 2 and Trigonometry

Question

In \(6-12,\) tell whether the variables vary directly, inversely, or neither. Each day, Sophia works for \(h\) hours typing \(p\) pages of a report at a rate of 15 minutes per page.

Step-by-Step Solution

Answer

The variables vary directly.

Step 1: Understand the Relationship

Sophia types pages at a rate of 15 minutes per page. If we need to find out how many pages she types in an hour, we convert the time to hours since 1 hour = 60 minutes. Therefore, she types \( \frac{60}{15} = 4 \) pages per hour.

Step 2: Define Variables and Equation

Consider the number of hours Sophia works as \( h \) and the number of pages she types as \( p \). From Step 1, we determined she can type 4 pages per hour. Therefore, the relationship can be expressed as \( p = 4h \).

Step 3: Determine the Type of Variation

The equation \( p = 4h \) suggests that as \( h \) increases, \( p \) also increases proportionally. This is a constant rate, indicating a direct relationship between \( p \) and \( h \) because they vary proportionally by a constant factor.

Key Concepts

Exploring the Relationship Between Variables

In the context of direct variation, understanding the relationship between variables is crucial. For instance, when Sophia works on typing pages, we look at the number of hours she works, denoted as \(h\), and compare it to the number of pages she completes, denoted as \(p\). These variables have a specific kind of relationship called direct variation.

• Direct Variation: This means that if one variable increases, the other does so as well. In Sophia's case, the more hours \(h\) she puts in, the more pages \(p\) she will type.
• Proportional Increase: Since the work Sophia does is per hour (4 pages per hour), as \(h\) goes up, \(p\) follows directly in proportion.

In simple terms, a direct relationship tells us that these variables move in tandem in a predictable way. The equation that shows this relationship is \(p = 4h\), indicating that every additional hour equals four more pages typed.

Understanding the Proportionality Constant

A proportionality constant is a fixed number that describes how two variables relate to each other in a direct variation. For Sophia's typing task, the proportionality constant tells us how efficiently or quickly she works.

• The Specific Constant: In the formula \(p = 4h\), the number 4 is the proportionality constant. This suggests that for every one unit increase in hours, the output grows by four pages.
• Significance:** The constant gives us a measure of the rate. It's like a rate of exchange between time and output, making predictions possible. If Sophia plans to work for 2 hours, she will type 8 pages, calculated simply by multiplying \(2 \times 4\).

The proportionality constant helps in creating clear expectations, ensuring one can predict how changing the hours spent will change the number of pages completed without having to compute from scratch each time.

Modeling with Mathematics

Mathematical modeling takes real-world scenarios and breaks them down into mathematical expressions or equations that are easier to analyze. Sophia's task of typing reports is essentially turned into a mathematical model with the equation \(p = 4h\).

• Why Model: Modeling helps in understanding, predicting, and strategizing outputs based on given inputs. For Sophia, the model gives her a means to predict how long she must work to achieve her goal of typing a certain number of pages.
• Benefit of a Model: It provides a clear framework; if Sophia needs to type 20 pages, she can divide 20 by 4, giving 5 hours as the working period required. This quick calculation showcases the practical value of mathematical modeling.

In summary, mathematical models simplify complex tasks, making planning and operational efficiency improvements straightforward through clear, easy-to-follow equations.