To analyze the given measurement of $37.0\overline{0}$ tons:

(a) Precision

The measurement is precise to the nearest hundred-thousandth of a ton. This is because the notation $0\overline{0}$ suggests repeating zeros after the decimal point, indicating precision to the fifth decimal place.

(b) Accuracy

Accuracy refers to how close a measured value is to the true value. Based on the information given, we can't determine the true value or how accurate this measurement is. If we assume the measurement is accurate, it would mean the measured value closely represents the actual weight.

(c) Greatest Possible Error (GPE)

The greatest possible error is half of the precision unit. Since the precision is to the nearest hundred-thousandth of a ton, the greatest possible error is:

$\text{GPE} = \frac{1}{2} \times 0.00001 = 0.000005 \text{ tons}$

This is the maximum amount by which the measured value might differ from the actual value.

Would you like further details or have any questions on this? Here are some related questions for practice:

1. What is the precision of the measurement $0.045$ meters?
2. How would you calculate the greatest possible error for a measurement precise to the nearest hundredth?
3. What is the difference between accuracy and precision in measurement?
4. If a measurement is given as $5.200$ liters, to what unit is it precise?
5. How would you express the greatest possible error for a weight measurement of $0.15$ kg?

Tip: Remember, precision refers to the smallest unit of measurement that a tool can measure, while accuracy reflects how close a measurement is to the true value.