Calculate Jonathan's Average Speed: Varying Speeds Over 3-Hour Cycling Competition

Jonathan is reviewing his cycling records from his last competition.

During the first half hour, he rode at a speed of 28 km/h.

The following two hours, he rode at a speed of 24 km/h, then 15 minutes downhill at a speed of 32 km/h, before continuing for another hour at a speed of 27 km/h.

What was his average speed?

To solve this problem, we'll follow these steps:

• Step 1: Calculate the distance for each segment.
• Step 2: Sum all distances for the total distance.
• Step 3: Sum all time intervals for the total time.
• Step 4: Use the average speed formula to find the solution.

Now, let's work through each step:
Step 1: Calculate the distance for each segment.
- First 30 minutes (0.5 hours) at 28 km/h: $\text{Distance} = 28 \times 0.5 = 14$ km.
- Next 2 hours at 24 km/h: $\text{Distance} = 24 \times 2 = 48$ km.
- Next 15 minutes (0.25 hours) at 32 km/h: $\text{Distance} = 32 \times 0.25 = 8$ km.
- Final 1 hour at 27 km/h: $\text{Distance} = 27 \times 1 = 27$ km.

Step 2: Total distance:
$\text{Total Distance} = 14 + 48 + 8 + 27 = 97$ km.

Step 3: Total time:
$\text{Total Time} = 0.5 + 2 + 0.25 + 1 = 3.75$ hours.

Step 4: Calculate the average speed:
$\text{Average Speed} = \frac{97}{3.75} \approx 25.888...$ km/h.

Therefore, the solution to the problem is 25.888....