What is the area, and approximate uncertainty in a circle....

[JustynSC]

Homework Statement

What is the area, and approximate uncertainty in a circle with radius 3.1*10^4 cm (or written: 3.1e4 cm)?

Homework Equations

Area=Pi*r^2

The Attempt at a Solution

My attempt to the solution took some trial and error, and it went as follows:
Substitute the circle's radius into the equation for the area of the circle: A=Pi(3.1e4)^2
Then I squared the () : A=Pi(9.61e8)
Following this I Multiplied by Pi: 3.017e9 cm^2 (sig fig) ==> 3.0e9 cm^2

This answer above Is correct, but in the book the answer is 3.0+/-0.2e9 cm^2.
The part that I do not understand is that the uncertainty they predict is +/-0.2

I figured that the uncertainty should be +/- 0.1e4, giving the radius a minimum of 3.0e4, and a max of 3.2e4.
Based on my way of working though the problem, my answer come out to be A=3.0+/-0.1e9 cm^2

If anyone can explain why my uncertainty isn't correct, and how they get that answer that would be wonderful!

 

[haruspex]

I agree with your answer, but the error in 3.1 is +/-0.05, not +/-0.1. That's a 1 in 60 error, so the error in the area should be 1 in 30.

 

[JustynSC]

I agree with your answer, but the error in 3.1 is +/-0.05, not +/-0.1. That's a 1 in 60 error, so the error in the area should be 1 in 30.

How do you come up with the error as 0.05 when the device used to measure does not specify that is has the exact measurements to the hundredth of a cm? In other words, if the .00 is not represented, how can it be used as the error?
Thirdly, I do not understand what you mean by 1 in 60 and 1 in 30

 

[haruspex]

How do you come up with the error as 0.05 when the device used to measure does not specify that is has the exact measurements to the hundredth of a cm? In other words, if the .00 is not represented, how can it be used as the error?
Thirdly, I do not understand what you mean by 1 in 60 and 1 in 30

You did not specify an error range for the radius, so it is implied by the number of significant digits. Quoting 3.1 implies something from 3.05 to 3.15. Had the radius been between 3.00 and 3.05?then the stated measurement should have been 3.0.

An error of 0.05 in a measurement of about 3 is a one part in sixty error, 0.05/3=1/60.

 

[JustynSC]

Thanks for the help! I get it now.

 

[JustynSC]

You did not specify an error range for the radius, so it is implied by the number of significant digits. Quoting 3.1 implies something from 3.05 to 3.15. Had the radius been between 3.00 and 3.05?then the stated measurement should have been 3.0.

An error of 0.05 in a measurement of about 3 is a one part in sixty error, 0.05/3=1/60.

So I was looking into the problem a little further and I might have missed a minor detail that explains why they did not use +/- 0.05 as the error. But I still can't figure out how they justify the answer to be within +/- 0.2
Here is the Note at the start of all problems:
assume a number like 6.4 is accurate to +/- 0.01, and 950 is +/-10 unless 950 is said to be "precisely" or "very nearly" 950, in which case assume 950 +/- 1.
I hope this help explain how they have their answer in the back of the book differing from ours.

 

[SammyS]

So I was looking into the problem a little further and I might have missed a minor detail that explains why they did not use +/- 0.05 as the error. But I still can't figure out how they justify the answer to be within +/- 0.2
Here is the Note at the start of all problems:
assume a number like 6.4 is accurate to +/- 0.01, and 950 is +/-10 unless 950 is said to be "precisely" or "very nearly" 950, in which case assume 950 +/- 1.
I hope this help explain how they have their answer in the back of the book differing from ours.

I believe you have a typo, and that the book says - or should have said - something like

assume a number like 6.4 is accurate to ± 0.1​

not ± 0.01.

 

[haruspex]

So I was looking into the problem a little further and I might have missed a minor detail that explains why they did not use +/- 0.05 as the error. But I still can't figure out how they justify the answer to be within +/- 0.2
Here is the Note at the start of all problems:
assume a number like 6.4 is accurate to +/- 0.01, and 950 is +/-10 unless 950 is said to be "precisely" or "very nearly" 950, in which case assume 950 +/- 1.
I hope this help explain how they have their answer in the back of the book differing from ours.

Assuming, as SammyS says, that should read 6.4+/-0.1...
Ok, but that is a bit unusual. One would normally take 6.4 as being accurate to that many figures, so represents a range 6.35 to 6.45.

giving the radius a minimum of 3.0e4, and a max of 3.2e4

Ok, but what areas do you calculate from those two radii?