# Significant figures problem

• sp3sp2sp
In summary, when calculating the areas of an unmeasured regulation soccer field, the least number of significant figures determines the number of significant figures in the answer. Between 100 and 110 means that the value could be 100, 101, 102, 103, 104, 105, 106, 107, 108, 109, or 110.f

## Homework Statement

A regulation soccer field for international play is a rectangle with a length between 100 m and 110 m and a width between 64 m and 75 m.
When calculating the areas what is the appropriate number of significant figures?

## Homework Equations

multiplying/dividing: The least number of significant figures in any number of the problem determines the number of significant figures in the answer.
adding/subtracting: final answer may have no more significant figures to the right of the decimal than the least number of significant figures in any number in the problem.

## The Attempt at a Solution

The thing that is confusing me is what exactly "between 100 and 110" means with respect to uncertainty and sig-figs.
I know that 100 has 1 sig-fig and that 110 has 2 sig-figs and that 64 and 75 both have 2 sig-figs.
I understand that between 100 and 110 means value could be 100,101,102,103,104,105,106,107,108,109 or 110.
100 has1 sig-fig, 110 has 2 sig-figs, and 101,102,103,104,105,106,107,108, and 109 have 3 sig-figs.
Need some help thanks

I know that 100
Then you have learned wrong. In cases such as 100, the number of significant digits is not clear. It can be one, two, or three.

• scottdave
From a practical point of view, having only 1 significant figure is not useful. This suggests that 100 could mean anything between 50 and 149? I'm not sure what your class teaches but go back and ask. ItsI good to know that.

100 has1 sig-fig, 110 has 2 sig-figs, and 101,102,103,104,105,106,107,108, and 109 have 3 sig-figs.
Need some help thanks
To elaborate on @Orodruin's remark: This sort of ambiguity is removed when numbers are correctly quoted as powers of 10.
1 × 102 has 1 significant figure, 1.0 × 102 has 2 and 1.00 × 102 has 3. Quoting "100" and then wondering about sig figs is meaningless.

I know that 100 has 1 sig-fig
In the context of this problem, the 100 is given as one of the error bounds. There is typically little need to concern oneself with the uncertainty in an error bound. One is concerned with the uncertainty in the true value given the measured result. That uncertainty is already explicitly stated. The true length is "between 100m and 110m". Yes, in the real world, the reported uncertainties are themselves uncertain. But when reporting results, the most important thing is the uncertainty in the measurement, not the uncertainty in the uncertainty. You can't report everything.

Given the ranges in which the true lengths and widths fall, what is the range within which the true area must fall? How large could the area possibly be? How small?

In the context of this problem, the 100 is given as one of the error bounds
I do not think it is useful to think of 100 as an error bound or a lower uncertainty limit in a measurement. It is a lower bound on the length of the field allowed by the rules. That in no way implies that it is related to any sort of statistical variance, it is just what is in the rules. You could aim to make all fields 101 m and that would be perfectly within the rules. A football field can typically be measured with much better precision than 10 m.

• SammyS
I do not think it is useful to think of 100 as an error bound or a lower uncertainty limit in a measurement. It is a lower bound on the length of the field allowed by the rules. That in no way implies that it is related to any sort of statistical variance, it is just what is in the rules. You could aim to make all fields 101 m and that would be perfectly within the rules. A football field can typically be measured with much better precision than 10 m.
My take is that we have been asked for the number of significant figures in the area of an unmeasured regulation field.

My take is that we have been asked for the number of significant figures in the area of an unmeasured regulation field.
The mention of "areas" (plural) suggests to me that the problem is to compute the minimal and maximal allowed areas, not any sort of mean value (which is impossible to know from the rules only, you would need a distribution).

• jbriggs444