To find the diameter of Mercury given the distance halfway around it, we can use the relationship between circumference (C) and diameter (D), which is given by the formula:
\[ C = \pi D \]
If the distance halfway around Mercury is 4,761 miles, then the full circumference (C) would be:
\[ C = 2 \times 4,761 = 9,522 \text{ miles} \]
Now we can rearrange the circumference formula to solve for the diameter:
\[ D = \frac{C}{\pi} \]
Now, substituting the circumference we found:
\[ D = \frac{9,522}{\pi} \]
Using \( \pi \approx 3.14159 \):
\[ D \approx \frac{9,522}{3.14159} \approx 3,032 \text{ miles} \]
Rounding to the nearest mile gives us:
\[ D \approx 3,031 \text{ miles} \]
Therefore, the length of the diameter of Mercury to the nearest mile is:
3,031 mi.