Problem 23 - Algebra Form and Function

Question

Abby and Leah go on a 5 hour drive for 325 miles at 65 mph. After \(t\) hours, Abby calculates the distance remaining by subtracting \(65 t\) from \(325,\) whereas Leah subtracts \(t\) from 5 then multiplies by \(65 .\) (a) Write expressions for each calculation. (b) Do the expressions in (a) define the same function?

Step-by-Step Solution

Answer

Explain your answer by referring to their expressions. Answer: Yes, Abby and Leah's calculations define the same function, as both of their expressions for the remaining distance are equal: \(D_A = D_L = 325 - 65t\). Abby calculates the remaining distance by subtracting 65t from 325, while Leah calculates by subtracting t from 5 and then multiplying the result by 65. After simplifying Leah's expression, we find that both expressions are the same, which means they define the same function.

Step 1: Abby's Calculation Method

Abby calculates the distance remaining by subtracting 65t (65 times the number of hours) from 325. We can write this as an expression: \(D_A = 325 - 65t\) Here, \(D_A\) represents the remaining distance according to Abby's method and \(t\) is the number of hours driven.

Step 2: Leah's Calculation Method

Leah calculates the remaining distance by first subtracting \(t\) (the number of hours) from 5 and then multiplying the result by 65. We can write this as an expression: \(D_L = 65(5 - t)\) Here, \(D_L\) represents the remaining distance according to Leah's method.

Step 3: Comparing the Functions

Now, let's see if the expressions define the same function by simplifying Leah's expression: \(D_L = 65(5 - t) = 65 \cdot 5 - 65t = 325 - 65t\)

Step 4: Conclusion

We can see that both expressions for the remaining distance are the same: \(D_A = D_L = 325 - 65t\) So, both Abby and Leah's calculations define the same function, as they are equal.

Key Concepts

Functions

In the context of this exercise, a function represents a relationship or rule that assigns a unique output value for each input value. The exercise involves determining if Abby and Leah's different calculations conceptually represent the same function. A function can be illustrated as a machine that takes an input, applies a specified rule, and produces an output.
Understanding function equivalence is crucial because different-looking mathematical expressions might actually represent the same function. Abby and Leah provide two distinct expressions to calculate the remaining distance of their trip:

• Abby uses: \( D_A = 325 - 65t \)
• Leah uses: \( D_L = 65(5 - t) \)

To determine if these expressions define the same function, we simplify and analyze their forms. The equivalent output for the same input value indicates the expressions define the same function.
Ultimately,

What Functions Teach Us

Functions define a systematic approach to solving problems. When different expressions define the same function, it showcases flexibility and creativity in mathematical problem-solving. This exercise helps build a deeper understanding of how functions work and how they are represented mathematically.

Expressions

An expression provides a mathematical phrase that can contain numbers, variables, and operators to represent a particular situation. In the exercise, Abby and Leah use expressions to calculate the distance remaining on their drive. It's important to understand what each part of these expressions represents.
Expressions aren't equations. They don't have an equals sign, but they can be used to construct equations. The expressions used by Abby and Leah illustrate how variables can interact with constants and coefficients:

• Abby's expression: \( 325 - 65t \) represents the remaining distance based on the total distance (325 miles) minus the miles traveled after \( t \) hours at 65 mph.
• Leah's expression: \( 65(5 - t) \) factors in the 5-hour total trip time, subtracting the hours driven \( t \), then multiplying by the speed (65 mph) to find the remaining distance.

Understanding expressions and their components—such as terms, coefficients, variables, and constants—is essential for building and simplifying expressions, which is a key skill in algebra.

Simplifying Expressions

Simplifying expressions is the process of making them easier to understand or solve. This involves combining like terms, factoring, and using the distributive property. Simplification can often reveal that seemingly different expressions are actually the same.
In the exercise, both Abby and Leah's expressions need simplification for comparison:

• Abby's method doesn’t need further simplification since it is already in its simplest form: \( 325 - 65t \).
• Leah’s expression \( 65(5 - t) \) needs simplification. By distributing the 65, it becomes \( 325 - 65t \), showing it is equivalent to Abby’s expression.

The Importance of Simplifying

Simplifying expressions lets us see the true nature of expressions without superficial differences. It aids in understanding algebraic structures and confirms equivalence between different forms. This skill is invaluable when solving equations or evaluating functions because it helps reduce complexity and identify underlying relationships.