Calculate the Average Speed: Turtle's Multi-Speed Journey to the Sea

To calculate the average speed of the turtle, we'll proceed with the following steps:

• Step 1: Convert 38 meters to kilometers, which gives us $0.038$ km.
  The first segment of 20 meters is $0.020$ km, and the second segment of 18 meters is $0.018$ km.
• Step 2: Calculate the time taken for the first segment.
  The speed is $X$ km/h for the first segment. The time is given by:
$t_1 = \frac{0.020}{X} \text{ hours}$
• Step 3: Calculate the time taken for the second segment.
  The speed for the second segment is $1.25X$ km/h. The time is:
$t_2 = \frac{0.018}{1.25X} \text{ hours}$
• Step 4: Convert the rest time of 20 minutes into hours, which yields $\frac{20}{60} = \frac{1}{3}$ hours.
• Step 5: Calculate the total time taken by adding the time for the two travel segments and the rest time:
$t_{\text{total}} = t_1 + \frac{1}{3} + t_2$ $t_{\text{total}} = \frac{0.020}{X} + \frac{1}{3} + \frac{0.018}{1.25X}$
• Step 6: Calculate the average speed using the formula for average speed, which is the total distance divided by the total time:
$\text{Average Speed} = \frac{0.038}{t_{\text{total}}}$
• Step 7: Simplify the expression:
  Calculate the times:
- $\frac{0.020}{X}$ hours - $\frac{0.018}{1.25X} = \frac{0.018}{\frac{5X}{4}} = \frac{0.018 \times 4}{5X} = \frac{0.072}{5X}$ - Total time = $\frac{0.020X + \frac{1}{3}(5X) + 0.072}{5X}$
• Step 8: Simplify the equation:
$\text{Total Time} = \frac{0.02 \times 5 + 1.666667 \times 5X + 0.072}{5X}$ $= \frac{0.1 + 1.666667X + 0.072}{5X}$ $= \frac{0.172 + 1.666667X}{5X}$
• Calculate the average speed:
$\text{Average Speed} = \frac{0.038}{\frac{0.172 + 1.666667X}{5X}} = \frac{0.038 \times 5X}{0.172 + 1.666667X}$ $= \frac{0.19X}{0.172 + 1.666667X}$

Thus, after calculation and simplification, the correct choice for the average speed is:

$\frac{142.5x}{129+1250x}$ km/h