Calculate Average Speed: Hugo's 7 Laps Around a 750-Meter Circular Track with Variable Times

To solve this problem, let's proceed with the following steps:

• Step 1: Calculate the circumference of the circular path.
  Since the diameter $d = 750$ meters, the circumference $C = \pi d = 750\pi$ meters.

• Step 2: Convert the circumference to kilometers:
  $C = \frac{750\pi}{1000} = 0.75\pi$ km.

• Step 3: Determine the total distance covered in 7 laps:
  Distance $d_{\text{total}} = 7 \times 0.75\pi = 5.25\pi$ km.

• Step 4: Calculate times for each set of laps in hours:
  - The first two laps in $10\frac{1}{2}$ minutes: $\frac{21}{2} \times \frac{1}{60} = \frac{21}{120} = \frac{7}{40}$ hours.
  - The next three laps at 9 km/h: Distance for these laps $= 3 \times 0.75\pi$, thus time $= \frac{3 \times 0.75\pi}{9} = \frac{0.75\pi}{3}$ hours.
  - The last two laps in 12 minutes: $\frac{12}{60} = \frac{1}{5}$ hours.

• Step 5: Calculate total time:
  Total time $t_{\text{total}} = \frac{7}{40} + \frac{0.75\pi}{3} + \frac{1}{5}$.

• Step 6: Simplify and calculate average speed:
  Using $\pi \approx 3.14$,
  $t_{\text{second}} = \frac{3 \times 0.75 \times 3.14}{9} \approx \frac{7.065}{9} \approx 0.785$ hours.
  Thus, $t_{\text{total}} = \frac{7}{40} + 0.785 + \frac{1}{5} \approx 0.175 + 0.785 + 0.2 = 1.16$ hours.
  Average speed $v_{\text{avg}} = \frac{5.25 \times 3.14}{1.16} \approx 10.732$ km/h.

Therefore, the average speed is $\boxed{10.732}$ km/h.