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The problem in the image describes a scenario where a vehicle uses different amounts of fuel while going uphill and downhill. You are tasked with finding the correct formula for the average fuel consumption rate (\(r_{\text{avg}}\)) when the distances traveled uphill and downhill are equal.

### Given information:
- The truck consumes 10 km per liter on level ground.
- It consumes 5 km per liter on an uphill path.
- It consumes 30 km per liter on a downhill path.

You are asked to prove that the total distance doesn't affect the average fuel consumption as long as the distances for uphill and downhill travel are the same.

#### The key to solving this problem:
- \(r_1 = \) the fuel consumption rate uphill (in km/liter).
- \(r_2 = \) the fuel consumption rate downhill (in km/liter).

The average fuel consumption rate for the trip is calculated using the harmonic mean formula when the distances are the same. The harmonic mean formula for two rates is given by:

\[
r_{\text{avg}} = \frac{2}{\frac{1}{r_1} + \frac{1}{r_2}}
\]

This formula ensures that the average fuel consumption takes into account the different rates of consumption uphill and downhill, which is the correct answer from the options provided.

The correct choice based on this reasoning is:

\[
r_{\text{avg}} = \frac{2}{\frac{1}{r_1} + \frac{1}{r_2}}
\]

This corresponds to the first option.

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Do you want more details on how the harmonic mean is derived, or have any other questions? Here are five related questions to deepen understanding:

1. How is the harmonic mean different from the arithmetic mean in fuel consumption scenarios?
2. Can the harmonic mean be used for more than two different rates of fuel consumption?
3. What would happen to the average fuel consumption rate if the distances traveled uphill and downhill were not equal?
4. How do varying fuel consumption rates affect overall vehicle efficiency?
5. What is the physical significance of using the harmonic mean in distance-related fuel problems?

**Tip:** The harmonic mean is particularly useful when dealing with rates, like speed or fuel consumption, where time or distance is the controlling factor.

Buktikan bahwa jarak tempuh pada lintasan menanjak dan menurun tidak akan memengaruhi rata-rata bahan bakar yang dihabiskan asalkan jarak keduanya sama. Rumus yang tepat untuk menyatakan rata-rata bahan bakar (r_avg) yang dihabiskan pada jarak yang sama adalah?

This problem explains how to calculate the average fuel consumption rate when traveling uphill and downhill, where the distances are equal. Using the harmonic mean formula for rates, this lesson is suitable for high school students studying rates and averages.

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