Problem 45 - Algebra Form and Function

Question

Find a formula for \(g\) by scaling the input and/or output of \(f\). Let \(f(t)\) give the measured precipitation in inches on day \(t\), and \(g(t)\) give the precipitation in centimeters. Use the fact that 1 in equals \(2.54 \mathrm{~cm}\).

Step-by-Step Solution

Answer

Question: Given the function \(f(t)\), which represents the precipitation in inches on day 't', find a formula for \(g(t)\), which represents the precipitation in centimeters on day 't', using the relationship that 1 inch equals 2.54 centimeters. Answer: To convert the precipitation in inches given by \(f(t)\) to centimeters represented by \(g(t)\), we can use the formula \(g(t) = 2.54 * f(t)\).

Step 1: Understand the relationship between inches and centimeters

We are given the relationship between inches and centimeters: 1 in equals 2.54 cm. To convert from inches to centimeters, we need to multiply the value in inches by 2.54.

Step 2: Define the function \(g(t)\)

Since g(t) represents the precipitation in centimeters on day 't', we need to convert f(t), the precipitation in inches on day 't', to centimeters. We'll do this by multiplying f(t) by the relationship between inches and centimeters.

Step 3: Formulate the formula for \(g(t)\)

To find the formula for \(g(t)\), we need to multiply \(f(t)\) by 2.54, the relationship between inches and centimeters. Therefore, the formula for \(g(t)\) is: \[g(t) = 2.54 * f(t)\] Now you have a formula to convert the precipitation in inches, given by \(f(t)\), to precipitation in centimeters, given by \(g(t)\).

Key Concepts

Understanding Functions

In the realm of mathematics, a function is a relationship between two distinct sets: the input and the output. When we talk about a function like \(f(t)\), we imply that it takes an input value \(t\) and produces an output, which in this exercise is the measured precipitation in inches. The key aspect of a function is its rule or formula that governs how the output is derived from the input. Functions are foundational in many mathematical applications because they provide a systematic way to transform one quantity into another. When modifying or scaling functions, like converting precipitation from inches \(f(t)\) to centimeters \(g(t)\), it's crucial to maintain the integrity of this relationship.

Measuring Precipitation

Precipitation measurement refers to quantifying the amount of rain, snow, sleet, or hail that falls over a specific period, usually measured in inches or centimeters. In this exercise, \(f(t)\) provides the precipitation in inches. Inch measurements are typically used in the United States. However, for scientific purposes and international consistency, the metric system is often preferred, with centimeters being the standard unit. To effectively communicate and compare precipitation data worldwide, it's crucial to convert measurements appropriately, which involves applying the conversion factor between inches and centimeters. This ensures that the data remains accurate and consistent across different measuring systems.

Role of Scaling Factors

Scaling factors are multiplicative coefficients used to convert one set of measurements into another. In the context of this exercise, the scaling factor serves as the bridge between the precipitation measurements in inches and centimeters. Given that one inch equals 2.54 centimeters, this factor—2.54—is the scaling factor. When transforming \(f(t)\) to \(g(t)\), each value

• is multiplied by 2.54.
• This adjustment scales up the measurement from inches to centimeters.
• Such conversions are vital in applications requiring standardized units for accuracy and clarity.

Scaling factors ensure that calculations and conversions between different units are seamless and precise, illustrating their importance in mathematical functions and real-world applications.