On a 20 -mile bike ride, you ride the first 10 miles at an...

On a 20 -mile bike ride, you ride the first 10 miles at an average speed of 8 $\mathrm{mi} / \mathrm{h}$ . What must your average speed over the next 10 miles be to have your average speed for the total 20 miles be (a) 4 $\mathrm{mi} / \mathrm{h} ?$ (b) 12 $\mathrm{mi} / \mathrm{h} ?$ (c) Given this average speed for the first 10 miles, can you possibly attain an average speed of 16 $\mathrm{mi} / \mathrm{h}$ for the total $20-$ mile ride? Explain.

Key Concepts

Average Speed Calculation

Average speed is defined as the total distance traveled divided by the total time taken to cover that distance. This concept emphasizes that the overall rate is influenced by the time spent at various speeds rather than simply averaging the speeds.

Time-Segment Relationship

When a journey is divided into segments, the time taken for each segment is determined by dividing the segment’s distance by the speed during that segment. The overall average speed of the trip is the total distance divided by the sum of the individual segment times, highlighting the importance of time in determining overall speed.

Weighted Averages and Harmonic Mean

For trips covering equal distances at different speeds, the overall average speed is not the arithmetic mean but is computed as the harmonic mean of the speeds. This concept shows that slower speeds have a greater impact on the overall average speed than faster speeds over equal distances.

Feasibility of Achieving a Target Average

This concept involves analyzing whether a particular target average speed can be met based on the time already spent. It requires solving for the necessary speed in remaining segments and checking if this calculated speed is physically or mathematically possible, acknowledging scenarios where the required speed might be infinite or unachievable.

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