the circumference of s sphere is measured to be 10 cm with an error of 0.4cm.the this radius is used to calculate the area and volume.find the errors in the calculated radius,area and volume
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 = 2πr, 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 / (2π).
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 / (2π) 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 = 4πr^2, and the volume (V) is given by V = (4/3)πr^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.