To find the errors in the calculated radius, area, and volume, we can use the concept of error propagation.
The circumference of a sphere is given by the formula C = 2Ï€r, where r is the radius of the sphere.
Given that the measured circumference is 10 cm with an error of 0.4 cm, we can express this as:
C = 10 cm ± 0.4 cm.
Let's start with finding the error in the radius (Δr). We can rearrange the circumference formula to solve for r:
r = C / (2Ï€).
Now, to calculate the error in the radius, we need to differentiate the formula with respect to C:
∂r/∂C = 1 / (2π).
Multiplying ∂r/∂C by ΔC (the error in C) gives us the error in the radius:
Δr = (∂r/∂C) * ΔC = (1 / (2π)) * 0.4 cm.
Therefore, the error in the calculated radius is 0.4 / (2Ï€) cm.
Next, let's find the error in the calculated area and volume of the sphere.
The surface area of a sphere (A) is given by the formula A = 4Ï€r^2, and the volume (V) is given by V = (4/3)Ï€r^3.
To find the errors in the area and volume, we'll differentiate these formulas with respect to r:
∂A/∂r = 8πr,
∂V/∂r = 4πr^2.
To calculate the error in the area, we'll multiply ∂A/∂r by Δr:
ΔA = (∂A/∂r) * Δr = (8πr) * (0.4 / (2π)) = 1.6r.
Similarly, to calculate the error in the volume, we'll multiply ∂V/∂r by Δr:
ΔV = (∂V/∂r) * Δr = (4πr^2) * (0.4 / (2π)) = 0.8r^2.
Therefore, the error in the calculated area is 1.6 times the calculated radius, and the error in the calculated volume is 0.8 times the square of the calculated radius.
Note: To obtain the specific numerical values for the errors, you need to substitute the measured value of the circumference into the equations and perform the calculations accordingly.