The radius of a circle is measured to be 12.8 ± 0.2 cm. I need to find the Area and the Circumference of the circle? But how to I account for the uncertainties in each value?
The radius has a relative uncertainty of 0.2/12.8 = ± 1.6%
The circumference will have the same relative uncertainty: 1.6% .
The relative uncertainty in the area will be twice as much: 3.2%, because area depends upon the square of radius.
In differential terms:
A = pi r^2
ln A = ln pi + 2 ln r
dA/A = 2 dr/r
The formula is good for small relative errors (less than about 10%)
To account for the uncertainties in each value, you can use error propagation formulas to find the uncertainties in the area and circumference of the circle. Here are the step-by-step calculations:
1. Calculate the area of the circle:
- Start with the formula for the area of a circle: A = πr^2
- Substitute the given radius value, r = 12.8 cm, into the formula: A = π(12.8 cm)^2
- Square the given uncertainty: (0.2 cm)^2 = 0.04 cm^2
- Calculate the area with the given radius: A = π(12.8 cm)^2 = 515.07264 cm^2
- Calculate the uncertainty in the area: ΔA = (2πr)Δr = 2π(12.8 cm)(0.2 cm) = 16.076 cm^2 (rounded to three significant figures)
- Final result: The area of the circle is 515.07264 cm^2 ± 16.076 cm^2.
2. Calculate the circumference of the circle:
- Use the formula for the circumference of a circle: C = 2πr
- Substitute the given radius value, r = 12.8 cm, into the formula: C = 2π(12.8 cm)
- Calculate the circumference with the given radius: C = 2π(12.8 cm) ≈ 80.425 cm
- Calculate the uncertainty in the circumference: ΔC = 2πΔr = 2π(0.2 cm) = 1.256 cm (rounded to three significant figures)
- Final result: The circumference of the circle is 80.425 cm ± 1.256 cm.
Therefore, the area of the circle is 515.07264 cm^2 ± 16.076 cm^2, and the circumference of the circle is 80.425 cm ± 1.256 cm.
To account for the uncertainties in the radius measurement, you will need to propagate the uncertainties to determine the uncertainties in the area and circumference of the circle. This process involves using the concept of error propagation.
1. Calculating the Area:
The formula to calculate the area of a circle is A = π * r^2, where r is the radius.
To account for the uncertainty in the radius, you need to propagate it to find the uncertainty in the area.
a) Calculate the maximum area:
Substituting the maximum radius value (12.8 + 0.2 cm) into the area formula:
A_max = π * (12.8 + 0.2)^2
b) Calculate the minimum area:
Substituting the minimum radius value (12.8 - 0.2 cm) into the area formula:
A_min = π * (12.8 - 0.2)^2
Now you have a range of values for the area: A_min to A_max.
2. Calculating the Circumference:
The formula to calculate the circumference of a circle is C = 2 * π * r.
Similar to the area, you will propagate the uncertainty in the radius to find the uncertainty in the circumference.
a) Calculate the maximum circumference:
Substituting the maximum radius value (12.8 + 0.2 cm) into the circumference formula:
C_max = 2 * π * (12.8 + 0.2)
b) Calculate the minimum circumference:
Substituting the minimum radius value (12.8 - 0.2 cm) into the circumference formula:
C_min = 2 * π * (12.8 - 0.2)
Now you have a range of values for the circumference: C_min to C_max.
Remember that the uncertainty range represents the possible deviation from the measured values due to the uncertainty in the radius measurement.