Problem 39 - College Algebra

Question

Use the four-step procedure for solving variation problems given on page 356 to solve. The time that it takes you to get to campus varies inversely as your driving rate. Averaging 20 miles per hour in terrible traffic, it takes you 1.5 hours to get to campus. How long would the trip take averaging 60 miles per hour?

Step-by-Step Solution

Answer

The trip would take 0.5 hours if the average speed were 60 miles per hour.

Step 1: Identify the type of variation and setup the equation

The problem is an example of inverse variation. So, setup the inverse variation equation as T = k/R, where T is time, R is driving rate and k is the constant of variation.

Step 2: Find the Constant of Variation

Insert the known values into the equation to find the constant of variation: 1.5 = k/20. Solving this equation for k gives k = 1.5 * 20 = 30.

Step 3: Use the Constant of Variation to answer the problem

Replace k in the original equation with the value we found (30) and solve for R = 60: T = 30/60 = 0.5.

Key Concepts

Variation Problems

Variation problems often involve understanding how one quantity changes in relation to another. These problems can be of various types, including direct variation, inverse variation, or joint variation. For direct variation, as one variable increases, the other also increases proportionally. Conversely, inverse variation, which is the focus here, means that as one variable increases, the other decreases. This is represented by the formula \( T = \frac{k}{R} \), where \(T\) represents the time it takes to travel, \(R\) is the driving rate, and \(k\) is the constant of variation.
Inverse variation problems can be identified by the relationship described in the problem statement, like the travel time decreasing as the speed increases.

• Always identify the type of variation from the problem context.
• Set up your equation reflecting this relationship.
• Use known values to find any unknown constants for easier calculations.

Understanding variation problems is critical in solving real-world issues where two quantities affect each other.

Constant of Variation

The constant of variation \(k\) is a key component in solving variation problems. It serves as the bridge that links two varying quantities. In the context of inverse variation, once you determine \(k\), you can calculate how the change in one quantity affects the other.
To find \(k\) in an inverse variation equation, you use the known values of the variables involved. For instance, if driving at 20 mph gets you to campus in 1.5 hours, then \(k\) is calculated as follows:
Insert these values into the equation you set up earlier, \(1.5 = \frac{k}{20}\). Solving for \(k\), we multiply both sides by 20, giving us \(k = 30\).

• Finding \(k\) is crucial because it remains constant as variables change.
• Use the constant to predict changes in other situations where the variables differ.

Grasping the concept of the constant of variation helps in swiftly solving different scenarios and ensuring consistency in calculations.

Driving Rate

The driving rate is essentially the speed at which you travel. In inverse variation problems like the one at hand, it directly influences the time taken to reach a destination.
As the problem states, when the driving rate is low, the travel time is higher and vice versa. This means if you average a higher speed, the time to reach the same place will reduce accordingly. For example, when increased from 20 mph to 60 mph, the time taken reduces because the driving rate is inversely proportional to time.

• Use the formula \(T = \frac{30}{R}\) after determining \(k\).
• Substitute different values of \(R\) to find the corresponding time \(T\).

Understanding the driving rate's effect in an inverse variation provides a practical insight into travel scenarios.