Problem 38 - Algebra Form and Function

Question

Find a formula for \(g\) by scaling the output of \(f\). Let \(f(t)\) give the area in square miles \(\left(\mathrm{mi}^{2}\right)\) of a town in year \(t\), and \(g(t)\) the area in square kilometers \(\left(\mathrm{km}^{2}\right.\) ). Use the fact that \(1 \mathrm{mi}^{2}\) equals \(2.59 \mathrm{~km}^{2}\).

Step-by-Step Solution

Answer

Answer: The formula for \(g(t)\) is \(g(t) = 2.59 \cdot f(t)\).

Step 1: Write down the given values

We are given the functions \(f(t)\) which represents the area in square miles, and we know that 1 square mile equals 2.59 square kilometers.

Step 2: Apply the conversion factor

To convert the output of \(f(t)\) from square miles to square kilometers, we need to multiply the output by the conversion factor 2.59. This means for any specific year \(t\), the area in square kilometers is \(2.59 \cdot f(t)\).

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

Now that we have the relation between the output of \(f(t)\) and the desired output for \(g(t)\), we can define the formula for \(g(t)\) as: \[g(t) = 2.59 \cdot f(t)\] This formula gives the area of the town in square kilometers at any year \(t\), by scaling the output of the original function \(f(t)\).

Key Concepts

Understanding Area Conversion

When we talk about area conversion in mathematics, we are referring to the process of changing measurements from one unit to another. This is important because different regions or countries may use various units for measuring land. In the example above, the area was initially measured in square miles, but needed to be converted to square kilometers.

Understanding area conversion involves knowing the specific conversion factor between the two units.

• For this exercise, the conversion factor is given as 1 square mile equals 2.59 square kilometers.

This means if you have an area expressed in square miles, you can multiply that by 2.59 to get the same area in square kilometers.

Performing conversions accurately is crucial, as it ensures consistency and clarity, especially in contexts like geography or planning, where different units are often compared.

Scaling Functions in Mathematics

Scaling a function in mathematics involves adjusting its output values by a fixed factor. This is like resizing all its estimates without changing the inputs or the nature of the function itself. Think of it as zoom-in or zoom-out for functions.

In our example, the function is scaled using a conversion factor. Here, the function \(f(t)\) outputs the area in square miles, and we needed to scale this by a factor of 2.59 to convert the outputs to square kilometers.

• The original function \(f(t)\) changes to \(g(t) = 2.59 \cdot f(t)\).

This scaling essentially transforms every value from \(f(t)\) to \(g(t)\) proportionally, maintaining the same relative discrepancies or differences between different years \(t\). This way, the overall shape and trend of the data remain intact, simply expressed in another unit.

Conversion Factors Explained

Conversion factors are the multipliers used to convert measurements from one unit to another. They are derived from the relationship between the two units. Having accurate conversion factors is essential for maintaining precision when changing units.

For example, a conversion factor like 2.59 between square miles and square kilometers means:

• 1 square mile will always equal 2.59 square kilometers.
• By multiplying the area in square miles by 2.59, you convert it to square kilometers.

It's a straightforward mathematical operation, yet critical in fields such as science, engineering, and logistics where unit conversions are a daily requirement. Never forget that a conversion factor acts as a bridge, allowing you to seamlessly transition between different measurement systems while ensuring that the physical quantity, in this case, the area, remains constant in essence.