Question

Find a formula for \(g\) by scaling the input and/or output of \(f\). Let \(f(t)\) give the number of acres lost to a wildfire \(t\) days after it is set, and \(g(n)\) give the number of hectares lost after \(n\) hours. Use the fact that 1 acre equals 0.405 hectares.

Short answer

Question: Given a function \(f(t)\) that gives the number of acres lost to a wildfire \(t\) days after it is set, derive a formula for \(g(n)\), which gives the number of hectares lost after \(n\) hours. Remember that 1 acre equals 0.405 hectares and there are 24 hours in a day. Answer: The formula for \(g(n)\) is \(g(n) = 0.405 \times f\left(\frac{n}{24}\right)\).

Step-by-step solution

Convert the input from days to hours

In order to convert the input from days to hours, we need to remember that there are 24 hours in a day. Therefore, if we want to know the number of acres lost after \(n\) hours, we need to evaluate the function \(f\) at \(\frac{n}{24}\) days: $$ f\left(\frac{n}{24}\right) $$ This gives us the number of acres lost after \(n\) hours.

Convert the output from acres to hectares

Now, we need to convert the output from acres to hectares using the conversion factor that 1 acre equals 0.405 hectares. To do this, simply multiply the output of the function in Step 1 by the conversion factor: $$ g(n) = 0.405 \times f\left(\frac{n}{24}\right) $$ Now, we have the formula for \(g(n)\), which gives the number of hectares lost after \(n\) hours. So, the formula for \(g(n)\) is $$ g(n) = 0.405 \times f\left(\frac{n}{24}\right) $$

Key concepts

Unit Conversion

When dealing with functions, unit conversion is a fundamental concept. It involves changing measurements from one unit to another, often for easier comparison or calculation. In this exercise, you see that the problem requires changing the measurement units from acres to hectares and from days to hours.

• For converting acres to hectares: As given, 1 acre equals 0.405 hectares. To handle the conversion in the function, multiply the number of acres by 0.405.
• For converting days to hours: There are 24 hours in a day. This means any measurement in days can be converted to hours by multiplying the days by 24. Conversely, to use hours in a day-based function like \(f(t)\), divide the number of hours by 24.

Unit conversion helps accurately measure and interpret the results regardless of the differing units involved, ensuring consistency and accuracy in the function's output.

Scaling

Scaling refers to adjusting the magnitude of input or output values in a function to achieve a desired effect or alignment. In this context, scaling is utilized to modify both the input and output of the function.

• When you scale the input by dividing \(n\) hours by 24, you're effectively rescaling the timeframe to match the function's original unit of days. This ensures that the function \(f(t)\) accurately represents the data over an hourly timeframe, rather than daily.
• For the output scaling by multiplying by 0.405, it adjusts the acres to hectares, thereby aligning the function's output to the correct unit, reflecting how many hectares were affected.

Scaling is a powerful tool for ensuring that functions are adaptable and correctly scaled to specific scenarios, whether by altering the timeframe or the units involved.

Input-Output Transformation

The concept of input-output transformation in functions involves modifying both the inputs and outputs to alter how a function behaves or represents real-world data. Here, transformation is necessary to convert the effects of the variables measured in one unit system to another.

• Input Transformation: Changing from days to hours means that you are transforming the input. This allows the function to be usable in different contexts or timeframes, broadening its applicability.
• Output Transformation: Converting acres to hectares involves transforming the output. By doing so, it provides the desired unit for interpretation while maintaining the integrity of the original data.

This process streamlines the ability of a single function to serve varied purposes or be compatible with different measurement units, embodying significant flexibility and utility.