Problem 33 - Algebra Form and Function

Question

Find a formula for \(g\) by scaling the output of \(f\). Let \(f(t)\) give the snowfall in feet \(t\) hours after a blizzard begins, and \(g(t)\) the snowfall in inches.

Step-by-Step Solution

Answer

Question: Given the function \(f(t)\) that represents the snowfall in feet, find a formula for \(g(t)\) which gives the snowfall in inches. Answer: The formula for \(g(t)\) is \(g(t) = 12 \cdot f(t)\).

Step 1: Identify the relationship between feet and inches

There are 12 inches in 1 foot, so the relationship between inches and feet can be expressed as: 1 foot = 12 inches

Step 2: Scale the output of \(f(t)\) to obtain \(g(t)\)

Since 1 foot = 12 inches, every foot of snow corresponds to 12 inches of snow. To find the snowfall in inches, we need to scale the output of \(f(t)\) by a factor of 12, which can be expressed in terms of the function \(g(t)\): \[g(t) = 12 \cdot f(t)\]

Step 3: Present the final formula for \(g(t)\)

Based on the scaling factor found in Step 2, the formula for \(g(t)\), which gives the snowfall in inches based on the snowfall in feet given by \(f(t)\), is as follows: \[g(t) = 12 \cdot f(t)\] This formula can be used to find the snowfall in inches at any given time \(t\) hours after a blizzard begins, based on the snowfall in feet provided by the function \(f(t)\).

Key Concepts

Understanding Scaling Function

Scaling a function is an important concept in mathematics that involves multiplying the output of a function by a constant to change its scale. In our example, the function that we are working with is about snowfall measurements. If we have function \(f(t)\) that outputs snowfall in feet, we can shift this to inches by scaling the function. Since there are 12 inches in one foot, the scaling factor here is 12. Thus, every time \(f(t)\) gives us a certain value in feet, multiplying this by 12 translates the measurement to inches. This is expressed in the formula:

• \(g(t) = 12 \cdot f(t)\)

The idea is to convert the measuring units of the entire function, ensuring that any output reflects a different, but equivalent value in inches based on the snowfall in feet. This systematic adjustment is crucial when dealing with different systems of measurement or units, allowing for seamless conversions.

Crafting a Function Formula

Formulating a function requires understanding the relationship between variables and how they correspond in different contexts. In this scenario, we have initially defined \(f(t)\) as the snowfall in feet for every \(t\) hours after a blizzard commences. To alter our perspective to inches, we need to derive a function formula that accounts for this change. The relationship between these two measurement types teaches us that if there is snowfall in feet described by \(f(t)\), the corresponding snowfall in inches is \(g(t)\). The function formula that accurately represents this transformation is:

• \(g(t) = 12 \cdot f(t)\)

This expression ensures that for any given hour \(t\), the function \(g\) reflects the snowfall in inches by considering the initial value in feet and directly multiplying it. Understanding how to develop such formulae is integral in using functions to represent real-world scenarios accurately.

Applying Units Conversion

Units conversion is a method that allows us to translate a quantity from one measurement system to another. This mathematical technique is essential, especially when working with functions that measure quantities like snowfall, distance, weight, etc. In the problem at hand, we start with a measurement in feet, where 1 foot equals 12 inches. To convert the function representing snowfall from feet to inches, apply the conversion factor of 12. This principle is central to the transformation:

• 1 foot = 12 inches
• \(g(t) = 12 \cdot f(t)\) conveys this unit change in a functional format

Using units conversion, we ensure consistency and precision when translating data to different measurement systems, making it easier to understand and apply in practical contexts. By applying this method systematically, we ensure that the data remains true to scale, without losing any of its original meaning or accuracy.