Problem 36 - Algebra Form and Function

Question

Find a formula for \(g\) by scaling the output of \(f\). Let \(f(t)\) give the speed in mph of a jet at time \(t\), and \(g(t)\) the speed in kilometers per hour (kph). Use the fact that \(1 \mathrm{kph}\) is \(0.621 \mathrm{mph}\).

Step-by-Step Solution

Answer

Question: If the speed of the jet is given by the function \(f(t)\) in miles per hour, find a formula for the speed of the jet in kilometers per hour, \(g(t)\), given that 1 kph equals 0.621 mph. Answer: \(g(t) = \frac{f(t)}{0.621}\)

Step 1: Write down the conversion factor between mph and kph.

We are given that 1 kph is equal to 0.621 mph. Let's write this conversion factor as a fraction: \(\frac{1 \ \text{kph}}{0.621 \ \text{mph}} = 1\).

Step 2: Set up an equation relating \(f(t)\) and \(g(t)\).

The objective is to find a formula for \(g(t)\) in terms of \(f(t)\). We know their respective units are in miles per hour and kilometers per hour, so let's relate their values using the conversion factor we found in Step 1: \(\frac{g(t)}{f(t)} = \frac{1 \ \text{kph}}{0.621 \ \text{mph}}\).

Step 3: Solve for \(g(t)\).

To solve for \(g(t)\), multiply both sides of the equation by \(f(t)\): \(g(t) = f(t) \cdot \frac{1 \ \text{kph}}{0.621 \ \text{mph}}\).

Step 4: Simplify the formula for \(g(t)\) using the conversion factor.

As we know that 1 kph is equal to 0.621 mph, we can replace the fraction on the right-hand side with the given conversion factor: \(g(t) = f(t) \cdot \frac{1}{0.621}\).

Step 5: Write the final formula for \(g(t)\) in terms of \(f(t)\) and the conversion factor.

Now that we have a formula for \(g(t)\), we can write it in terms of \(f(t)\) and the given conversion factor. Since we want \(g(t)\) in kilometers per hour, we need to divide \(f(t)\) by the conversion factor: \(g(t) = \frac{f(t)}{0.621}\). This is the final formula for \(g(t)\) in terms of \(f(t)\). The speed of the jet in kilometers per hour, \(g(t)\), can be found by dividing the speed in miles per hour, \(f(t)\), by the conversion factor 0.621.

Key Concepts

Understanding Rate of Change

The rate of change is a fundamental concept in mathematics, especially in calculus and algebra. It describes how one quantity changes with respect to another. In this exercise, the rate of change helps us understand how speed is changing over time.

• Rate of Change in Speed: When we calculate the speed of the jet in kilometers per hour instead of miles per hour, we are effectively determining how much the speed changes as we switch units.
• Interpreting Formula: The formula \( g(t) = \frac{f(t)}{0.621} \) signifies that for every 0.621 miles per hour, there's a corresponding 1 kilometer per hour. This conversion indicates a constant rate of change from mph to kph.

By grasping the concept of rate of change, you can better understand how different quantities relate in real-world situations.

Exploring Function Transformation

Function transformation is a process that changes a function's position, size, and other properties. In simple terms, it’s about altering the graph of a function.

• Scaling the Output: In our scenario, we are scaling the speed function. By using a conversion factor of \( \frac{1}{0.621} \), we transform the function \( f(t) \), expressed in mph, into \( g(t) \), expressed in kph.
• Graphical Interpretation: If \( f(t) \) were plotted on a graph, scaling it by \( \frac{1}{0.621} \) would stretch or compress the graph vertically. This means every mph unit on the vertical axis of the graph will now correspond to approximately 1.61 kph instead.

Understanding how function transformation works is key to modifying functions for different contexts and applications.

Crafting Algebraic Formulas

Algebraic formulas are structured expressions that represent relationships between variables.

• Building Formulas: The given problem asks us to construct an algebraic formula to express \( g(t) \) in terms of \( f(t) \). We use the conversion factor, known from the exercise, to relate these two quantities.
• Steps to Formulate: We started with the relation \( \frac{g(t)}{f(t)} = \frac{1 \ \text{kph}}{0.621 \ \text{mph}} \). Then, multiplying both sides by \( f(t) \), we rearrange it to find \( g(t) = \frac{f(t)}{0.621} \).
• Importance: By organizing a systematic approach to create this formula, we ensure we can correctly convert and relate various rates in mathematics, which is vital in fields like engineering and physics.

Mastering the skill of crafting algebraic formulas is essential for solving real-life problems that involve mathematical relationships.