The circumference of a sphere was measured to be cm with a possible error of cm. Use linear approximation to estimate the maximum error in the calculated surface area.
I see that you posted this again with actual numbers. See my later answer.
To estimate the maximum error in the calculated surface area of the sphere, we can use linear approximation.
Let's start with the formula for the surface area of a sphere:
Surface Area = 4πr^2
First, we need to find the radius (r) of the sphere. We can do this by using the formula for the circumference of a sphere:
Circumference = 2πr
Rearranging this equation, we have:
r = Circumference / (2π)
Substituting the given values into the equation, we have:
r = cm / (2π)
Now, let's find the maximum error in the radius. The possible error in the circumference is given as cm. Since the radius is calculated based on the circumference, the error in the radius will be the same as the error in the circumference.
Therefore, the maximum error in the radius (δr) is cm.
Next, let's find the maximum error in the surface area (δA). We can use linear approximation to estimate this error.
The formula for the linear approximation error is given by:
δA = 2πrδr
Substituting the values we found earlier, we have:
δA = 2π * (cm / (2π)) * cm
Simplifying the expression, we have:
δA = cm^2
Thus, the maximum error in the calculated surface area of the sphere is cm^2