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Chapter 0: Problem 92

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.

Short Answer

Expert verified
The simplified expression for the average rate on a round-trip commute is \( \frac{2}{\frac{1}{r_{1}}+\frac{1}{r_{2}}} \). For the given rates of 40 mph (outgoing) and 30 mph (returning), the average rate is approximately 34.29 mph. The answer is not 35 mph because the average speed of a round trip is not simply the average of the speeds on the individual trips.

Step by step solution

01

Simplify the Complex Rational Expression

First, take the complex rational expression \( \frac{2 d}{\frac{d}{r_{1}}+\frac{d}{r_{2}}} \) and simplify it. Notice that the distance, \( d \), appears in the numerator and denominator of the expression. Factor the \(d\) out of the denominator and cancel out with the numerator to get \( \frac{2}{\frac{1}{r_{1}}+\frac{1}{r_{2}}} \) .
02

Substitute the given average rates

Substitute the given values \( r_{1} = 40 \) mph (miles per hour) and \( r_{2} = 30 \) mph into the simplified expression to calculate the average rate: \( \frac{2}{\frac{1}{40}+\frac{1}{30}} \) .
03

Calculate the Average Rate

Perform the calculation to get the average rate for the round trip commute. The result is approximately 34.29 mph.
04

Explain the result

Speed is a ratio of distance and time. The average speed for the entire round trip is not simply the average of the initial and returning speed. This is because the time taken for each trip is different. With a lower speed, the return trip takes longer time. Therefore, the average speed for the round trip is less than 35 mph, which would be the simple average of the individual speed values.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Average Rate Calculation
Understanding how to calculate the average rate helps in scenarios where speeds vary over sections of a journey. The average rate formula provides a comprehensive picture of the speed over the complete trip rather than at individual segments. The formula is useful because it takes into account both the distance and the time for each segment.
For any round trip, find the average speed using the formula:
  • \[ \frac{2}{\frac{1}{r_{1}} + \frac{1}{r_{2}}} \]
Once you substitute and solve for the given speeds, this formula gives a more accurate average speed.
It's essential because simply averaging the speeds doesn't account for the different times spent at each speed.
Round-Trip Speed
Round-trip speed differs from regular speed calculation. When you commute a round-trip, each leg of the journey can have different speeds. The formula for round-trip speed considers these variations.
If you drive to a destination at a certain speed and return at another speed, the time spent traveling at each speed is critical. For instance:
  • Driving to your destination at 40 mph.
  • Returning at 30 mph.
The average speed is not just the numerical average of 40 and 30, which is 35 mph. It accounts for how long you spend traveling at each speed.
Since more time is spent traveling at the slower speed, the overall average is pulled down below the simple average (35 mph in this scenario). Calculating it correctly gives you a true representation of your round-trip speed.
Simplifying Expressions
The key to simplifying expressions, especially complex rational ones, lies in recognizing common factors and terms. Start by identifying parts of the expression that can be simplified.
The given expression is:
  • \[ \frac{2d}{\frac{d}{r_{1}} + \frac{d}{r_{2}}} \]
Notice the variable \(d\) on both the top and bottom. By factoring \(d\) out and canceling it out, you simplify the expression down to:
  • \[ \frac{2}{\frac{1}{r_{1}} + \frac{1}{r_{2}}} \]
This formula now focuses on rates \(r_{1}\) and \(r_{2}\), making the calculation clearer and more concise.
Simplifying expressions effectively saves you from unnecessary calculations and provides clarity in solving complex problems.

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Most popular questions from this chapter

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