Question

Find a formula for \(g\) by scaling the input and/or output of \(f\). . Let \(f(t)\) give the area in acres of a county's topsoil lost to erosion in year \(t,\) and \(g(t)\) give the area in hectares lost. Use the fact that 1 acre equals 0.405 hectares.

Short answer

Question: Given that the function \(f(t)\) represents the area of lost topsoil in acres, find the function \(g(t)\) that represents the area of lost topsoil in hectares, using the conversion factor that 1 acre equals 0.405 hectares. Answer: \(g(t) = 0.405f(t)\)

Step-by-step solution

Understand the relationship between acres and hectares

Since 1 acre equals 0.405 hectares, we can convert acres to hectares by multiplying the number of acres by 0.405. We can use this fact in finding the formula for \(g(t)\) by scaling the output of \(f(t)\).

Scale the output of \(f(t)\)

To find \(g(t)\), we need to scale the output of \(f(t)\) by multiplying it with the conversion factor 0.405. This means that for every unit increase in the value of \(f(t)\), the output of \(g(t)\) should also increase by 0.405 times. Therefore, we can write the formula for \(g(t)\) as: \[g(t) = 0.405f(t)\] Now, we have the formula for \(g(t)\) in terms of \(f(t)\), which will give us the area of lost topsoil in hectares.

Key concepts

Algebraic Functions

Algebraic functions are mathematical expressions that use variables and operations to create equations. In this problem, we have two functions—\(f(t)\) and \(g(t)\). Each function represents an algebraic relationship between variables and their respective measurements in different units. Here, \(f(t)\) gives us the topsoil erosion area in acres, and \(g(t)\) provides it in hectares. Understanding and manipulating these algebraic functions allows us to convert between units systematically. By scaling the output of an algebraic function, we effectively manage to convert its results into a different measurement system. This transformation creates an equation beneficial for making direct comparisons across different units, exemplified in our equation \(g(t) = 0.405f(t)\). This process demonstrates the versatility of algebraic functions and their capacity to simplify complex real-world phenomena like topsoil erosion. Algebraic functions can thus be adapted or rewritten to reflect various probable scenarios, making them a powerful tool in mathematics.

Unit Conversion

Unit conversion is an important mathematical concept used to translate measurements from one system of units to another. In this particular exercise, we have the task of converting acreage, the area measured in acres, into hectares. To perform this conversion accurately, we need to recognize the relationship or conversion factor between the two units. We know that 1 acre is equal to 0.405 hectares. By multiplying the area in acres by 0.405, we create a direct conversion formula:

• Area in hectares = Area in acres × 0.405

This approach is applied to the function \(f(t)\), which calculates area loss in acres, yielding a new function, \(g(t)\), that reveals the same loss in hectares. By understanding and implementing unit conversion, we ensure that our mathematical solutions remain adaptable to various practical applications and contexts.

Problem Solving

Problem solving is an essential skill that combines various mathematical techniques to find solutions to complex issues. In this task, we're asked to develop a function \(g(t)\) from \(f(t)\) to express measurements in different units. Step-by-step problem solving begins with understanding the given problem and the relationship between the variables involved. First, recognize that we are dealing with unit conversion, which requires scaling the output from one unit to another using a conversion factor. This shows the necessity of correctly applying the 0.405 conversion factor to transform acres into hectares. We then apply problem solving methods like:

• Identifying given data and what is required.
• Establishing a clear understanding of the relationship and conversion factor involved.
• Using algebraic functions to represent this relationship mathematically.

By following these steps, we derive the formula \(g(t) = 0.405f(t)\), effectively solving the problem through structured mathematical reasoning.