Problem 22 - Algebra Form and Function

Question

If you drive to work at \(v\) miles per hour, the time available for breakfast is \(B(v)=30-480 / v\) minutes. (a) Which is greater, \(B(35)\) or \(B(45) ?\) (b) Explain your answer in terms of the expression for \(B(v)\) and give a practical interpretation.

Step-by-Step Solution

Answer

Answer: B(35) is greater than B(45). This means that driving at a speed of 35 miles per hour allows for more time for breakfast compared to driving at 45 miles per hour. The faster you drive, the less time you have for breakfast due to the reduced travel time.

Step 1: Calculate B(35)

To find the time available for breakfast when driving at 35 miles per hour, plug in \(v=35\) into the function: \(B(35) = 30 - \frac{480}{35}\). Now, calculate the result by performing the mathematical operations.

Step 2: Calculate B(45)

Similarly, to find the time available for breakfast when driving at 45 miles per hour, plug in \(v=45\) into the function: \(B(45) = 30 - \frac{480}{45}\). Now, calculate the result by performing the mathematical operations.

Step 3: Compare B(35) and B(45)

Now that we have computed B(35) and B(45), we compare the two results to determine which value is greater.

Step 4: Explain the result

Based on our findings in steps 1-3, explain the results in terms of the expression for B(v) and give a practical interpretation for why one value would be greater than the other.

Key Concepts

Algebraic Functions

Algebraic functions form the foundation of modern mathematics, and understanding them is crucial for problem-solving and interpreting mathematical models. In this context, the function \(B(v) = 30 - \frac{480}{v}\) describes a real-world scenario where the time available for breakfast depends on the speed \(v\) at which you drive to work. This is an example of a rational function, characterized by the division of a constant by a variable.In this function:

• \(30\) represents the total available time in minutes without any driving time considered.
• \(\frac{480}{v}\) represents the reduction in available time due to driving, where 480 is a constant factor determined by the problem context.

Understanding how algebraic functions like this one work enables students to model various real-life situations mathematically, leading to more effective decision-making.

Problem Solving

Problem-solving with algebraic functions involves multiple steps to find the desired results. Here, we solve for \(B(35)\) and \(B(45)\) to see which is greater. First, calculate \(B(35)\) by substituting 35 into the function:\[B(35) = 30 - \frac{480}{35} = 30 - 13.71 = 16.29\] This means you have approximately 16.29 minutes for breakfast.Next, tackle \(B(45)\):\[B(45) = 30 - \frac{480}{45} = 30 - 10.67 = 19.33\] This translates to about 19.33 minutes.Comparing these outcomes reveals that \(B(45)\) (19.33 minutes) is greater than \(B(35)\) (16.29 minutes). This exercise not only shows the steps involved in solving the problem but also emphasizes the importance of precise mathematical calculations.

Mathematical Interpretation

Interpreting the results mathematically provides insights into the behavior of the function and its real-world implications. The algebraic expression \(B(v) = 30 - \frac{480}{v}\) clearly shows that as the speed \(v\) increases, the time available for breakfast \(B(v)\) also increases. This is because:

• Higher speed means the denominator \(v\) of the fraction \(\frac{480}{v}\) becomes larger.
• A larger denominator results in a smaller fraction value, reducing the subtraction from 30.
• Thus, more breakfast time remains when you drive faster.

In practical terms, driving at a higher speed allows for more time for other activities before work, like having breakfast. This demonstrates how mathematical interpretations of functions can lead to better understanding and decision-making in everyday life scenarios.