Approximate uncertainty in area of circle

In summary, the approximate uncertainty in the area of a circle with a radius of 5.3 * 104 cm is about 4 * 108 cm2. This is based on an upper limit of 9.2 * 109 cm2 and a lower limit of 3 * 108 cm2. The difference in area can also be related to the difference in radius.
  • #1
chops369
56
0

Homework Statement


What is the approximate uncertainty in the area of a circle of radius 5.3 * 104 cm? Express your answer using one significant figure.


Homework Equations


A = pi*r2


The Attempt at a Solution


Using the given radius, I found the area to be 8.8 * 109 cm2.

And since the uncertainty is not given, I'm assuming that it's 0.1 * 104 cm.

Using this, the upper limit for the radius is 5.4 * 104 cm, which makes the upper limit for the area 9.2 * 109 cm2.

Subtracting 9.2 - 8.8 = 0.4 * 109 cm2. But this is apparently not the correct answer, and I can't figure out why. Unless I'm forgetting some fundamental aspect of significant figures, the only other way I can express this in one sig fig is to write out the actual number, i.e. 400000000; but this is also incorrect.

What am I doing wrong?
 
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  • #2
If the area is A = pi*r2, then the difference in area is delta-A (sorry, I forgot all my latex). Can you relate delta-A to delta-r (the radius)? If you've taken calculus, then instead of delta-A it would be dA/dr.
 
  • #3
I just figured it out.

Turns out it was a rounding error on my part, so what I rounded up to 4 * 108 should have actually been rounded down to 3 * 108.
 
  • #4
chops369 said:
Using this, the upper limit for the radius is 5.4 * 104 cm, which makes the upper limit for the area 9.2 * 109 cm2.

That's the upper bound, now what's the lower bound?
 
  • #5


As a scientist, it is important to understand and properly handle uncertainties in calculations. In this case, the uncertainty in the radius is given as 0.1 * 104 cm, which means the actual radius could be anywhere between 5.2 * 104 cm and 5.4 * 104 cm. Therefore, the upper limit for the area would be 9.2 * 109 cm2 and the lower limit would be 8.4 * 109 cm2. To express this uncertainty in one significant figure, we need to round the difference between these two values to one significant figure, which would be 1 * 109 cm2. Therefore, the approximate uncertainty in the area of the circle is 1 * 109 cm2. This means that the area of the circle with a radius of 5.3 * 104 cm could be anywhere between 8.4 * 109 cm2 and 9.4 * 109 cm2, with an uncertainty of 1 * 109 cm2. This is the correct way to express the uncertainty in the area of the circle.
 

1. What is approximate uncertainty in the area of a circle?

Approximate uncertainty in the area of a circle refers to the amount of error or variability in the calculated value of the area of a circle. It takes into account the limitations of measurement and other factors that may affect the accuracy of the calculation.

2. How is approximate uncertainty in the area of a circle calculated?

Approximate uncertainty in the area of a circle is typically calculated using the formula for uncertainty propagation, which takes into account the uncertainties in the measurements of the radius or diameter of the circle.

3. What factors can affect the approximate uncertainty in the area of a circle?

Factors that can affect the approximate uncertainty in the area of a circle include the accuracy of the measuring tools used, the precision of the measurements, and any potential sources of error in the calculation process.

4. How can approximate uncertainty in the area of a circle be minimized?

To minimize the approximate uncertainty in the area of a circle, it is important to use precise and accurate measuring tools and techniques, and to carefully consider and account for any potential sources of error in the calculation process.

5. Why is it important to consider approximate uncertainty in the area of a circle?

Considering approximate uncertainty in the area of a circle is important because it provides important information about the reliability and accuracy of the calculated value. It also allows for a better understanding of the potential range of values that the area could fall within, providing a more complete picture of the data.

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