Problem 42 - Algebra Form and Function

Question

You drive 100 miles. Over the first 50 miles you drive \(50 \mathrm{mph}\), and over the second 50 miles you drive \(V\) mph. (a) Calculate the time spent on the first 50 miles and on the second 50 miles. (b) Calculate the average speed for the entire 100 mile journey. (c) If you want to average 75 mph for the entire journey, what is \(V ?\)

Step-by-Step Solution

Answer

Answer: 150 mph

Step 1: (a) Calculate the time spent on the first 50 miles and the second 50 miles

To calculate the time spent on each part of the journey, we will use the formula Time = Distance / Speed. For the first 50 miles: Distance = 50 miles Speed = 50 mph Time = Distance / Speed Time = 50 miles / 50 mph = 1 hour For the second 50 miles: Distance = 50 miles Speed = V mph Time = Distance / Speed Time = 50 miles / V mph The times are 1 hour for the first part and (50/V) hours for the second part.

Step 2: (b) Calculate the average speed for the entire 100-mile journey

To calculate the average speed, we will use the formula Average Speed = Total Distance / Total Time. Total Distance = 100 miles Total Time = Time for first 50 miles + Time for the second 50 miles Total Time = 1 hour + (50/V) hours Average Speed = Total Distance / Total Time Average Speed = 100 miles / (1 + 50/V) hours

Step 3: (c) Find the value of V to have an average speed of 75 mph for the entire journey

We want the average speed to be 75 mph, so we will set the Average Speed equation equal to 75 and solve for V. 75 mph = 100 miles / (1 + 50/V) hours To solve for V, we can follow these steps: 1. Multiply both sides of the equation by (1 + 50/V) to remove it from the denominator: 75(1 + 50/V) = 100 2. Distribute the 75 to both terms in the parentheses: 75 + 3750/V = 100 3. Subtract 75 from both sides to isolate the term with V: 3750/V = 25 4. Multiply both sides by V to eliminate the fraction: 3750 = 25V 5. Divide by 25 to solve for V: V = 150 So, to average 75 mph for the entire journey, you need to drive 150 mph for the second part of the trip.

Key Concepts

Understanding the Distance-Time Relationship

The distance-time relationship is a fundamental concept when analyzing motion. It tells us how long it takes to travel a certain distance at a given speed. Understanding this relationship lets us calculate how much time we'll need, or how far we can go, given a specific speed. It revolves around a simple formula:

• Time = Distance / Speed

Let's break it down with an example. Imagine driving the first 50 miles of your journey at 50 mph. Using our formula:

• Time = 50 miles / 50 mph = 1 hour

This means it will take you 1 hour to cover 50 miles at this speed. Now, if you were to drive another segment with a different speed, such as driving the second half at an unknown speed \(V\), the time taken would be:

• Time = 50 miles / V mph = 50/V hours

Understanding this relationship helps in planning and managing travel time effectively.

How to Calculate Average Speed

Average speed is essential when you want to know the overall efficiency of a trip. It's not just the speed at a given moment, but rather the total distance divided by the total time taken. For instance, let's look at a 100-mile journey where the first half is driven at 50 mph. The time for the first half is 1 hour. For the second half, if driven at speed \(V\), the time is \(50/V\) hours. Added together, the total journey time is:

• Total Time = 1 + \(50/V\) hours

Then, to find the average speed for the entire trip, we use:

• Average Speed = Total Distance / Total Time

For a 100-mile journey this results in:

• Average Speed = 100 miles / (1 + 50/V) hours

This formula helps assess overall travel speed and can guide adjustments in parts of the trip to meet specific goals.

Solving Problems with Algebraic Equations

Algebraic equations are powerful tools for solving for unknowns based on given conditions. In this context, they help us find the speed required to meet a travel goal.Suppose you want your average speed for a whole trip to be 75 mph. You'd set up your equation by equating average speed to 75 and solving for the unknown speed \(V\):

• \(75 = 100 / (1 + 50/V)\)

To solve this, follow these steps:1. **Eliminate the Fraction.** Multiply both sides by \(1 + 50/V\): \\[ 75(1 + 50/V) = 100 \]2. **Distribute and Simplify.** Expand and solve the equation:

• 75 + 3750/V = 100

3. **Isolate V.** Subtract 75 from 100:

• 3750/V = 25

4. **Solve for V:**

• 3750 = 25V
• V = 150

The conclusion is you must drive 150 mph during the second half to average 75 mph overall. It illustrates how precise calculations using algebraic operations can achieve specific travel goals.