Problem 82 - Algebra and Trigonometry

Question

The average rate on a round-trip commute having a one-way distance d is given by the complex rational expression $$ \frac{2 d}{\frac{d}{r_{1}}+\frac{d}{r_{2}}} $$ in which \(r_{1}\) and \(r_{2}\) are the average rates on the outgoing and return trips respectively. Simplify the expression. Then find your average rate if you drive to campus averaging 40 miles per hour and return home on the same route averaging 30 miles per hour. Explain why the answer is not 35 miles per hour.

Step-by-Step Solution

Answer

The average speed for the entire journey is approximately 34.29 mph.

Step 1: Simplifying the Expression

The formula given is for the harmonic mean, which we will simplify. Rewrite the equation as: \[\frac{2 d}{\frac{d}{r_{1}}+\frac{d}{r_{2}}} = \frac{2 d}{d\left(\frac{1}{r_{1}}+\frac{1}{r_{2}}\right)}= \frac{2}{\left(\frac{1}{r_{1}}+\frac{1}{r_{2}}\right)}\]

Step 2: Substituting given rates

Now we'll substitute the given values for \(r_{1}\) and \(r_{2}\) into the simplified equation from Step 1. In this case, \(r_{1}\) and \(r_{2}\) are the average speeds on the outgoing and return trips, which are 40 mph and 30 mph respectively. So, replace \(r_{1}\) with 40 and \(r_{2}\) with 30 to get:\[\frac{2}{\left(\frac{1}{40}+\frac{1}{30}\right)}\]

Step 3: Calculating the Total Average Speed

Now we'll calculate the above expression to arrive at the actual average speed. Simplifying it gives an average speed of approximately 34.29 mph.

Step 4: Explanation why the Average Speed is not 35 mph

When we take the average of 40 mph and 30 mph, we would usually get 35 mph. However, in the case of traveling speeds, it must be considered that more time is spent at the slower speed, which will bring down the average.

Step 5: Conclusion

So, if a student drives to campus averaging 40 mph and returns home on the same route averaging 30 mph, the average speed for the entire journey will not be the average of the two speeds (35 mph), but instead approximately 34.29 mph. This is because of the greater amount of time spent at the slower speed, which lowers the average.

Key Concepts

Complex Rational Expression

Complex rational expressions may look intimidating, but they're quite manageable once broken down. Think of a complex rational expression as a fraction where the numerator or the denominator (or both) are themselves fractions. These appear complicated initially, but simplification is key.

In our problem, we encounter a complex rational expression:

• Numerator: \(2d\)
• Denominator: \(\frac{d}{r_{1}} + \frac{d}{r_{2}}\)

To simplify \(\frac{2d}{\frac{d}{r_{1}} + \frac{d}{r_{2}}}\), we multiply both the numerator and the denominator by the common denominator \(d\), turning it into \(\frac{2}{\left(\frac{1}{r_{1}}+\frac{1}{r_{2}}\right)}\).

This simplification process helps translate a complicated-looking expression into a far more manageable form, often used in scientific applications or in solving rate problems efficiently.

Average Rate

The term 'average rate' often stirs confusion, especially when applied to rates like speed. When dealing with average rates, particularly over a distance covered in different stretches of speed, the harmonic mean rather than the arithmetic mean is the right calculation.

Our example uses a complex expression to represent the harmonic mean to find the average speed during a round-trip journey. This is crucial, as the time spent traveling at each rate is different, so simply finding the arithmetic mean (adding the rates and dividing by two) would not account for these differences.

When simplifying \(\frac{2}{\left(\frac{1}{40} + \frac{1}{30}\right)}\), you find the correct average speed to be about 34.29 mph. Here, the harmonic mean correctly weights the speed based on the time spent at each rate, resulting in a more accurate average than the straightforward arithmetic mean would provide.

Speed

Speed is how fast something moves over a particular distance in a given time. In our scenario, we use certain speeds for different legs of a commute, namely 40 mph and 30 mph.

To find the average speed over different segments, simply averaging them isn't enough due to differences in time spent at each speed. This time discrepancy makes speeds different from distances when calculating averages. Our journey shows why: more time at 30 mph (a slower speed) means it affects the average speed more, pulling it down below 35 mph. The harmonic mean offers the solution by considering both speeds and the relative time spent traveling at those speeds.

Understanding this concept reinforces the necessity of using appropriate mathematical methods depending on what needs averaging in varied scenarios.

Distance

Distance plays a core role in computing speed and time relationships. In this exercise, the round-trip distance adds context to the trip's speed calculations. The total distance for a commute affects how we interpret time spent at different speeds.

Since the problem provides a one-way distance \(d\), the round trip covers the distance twice. Notably, the speed and distance components link through time, an essential factor in travel scenarios. More distance covered at a slower speed increases total trip time disproportionately, highlighting why the average speed calculation must be nuanced.

Knowing how distance connects to speed and time aids in fully appreciating why the harmonic mean and not a simple average dictates the average speed. This knowledge empowers students to tackle similar problems, equipped to analyze the relationship between these elements accurately.