The circumference of a sphere was measured to be 74.000 cm with a possible error of 0.50000 cm. Use linear approximation to estimate the maximum error in the calculated surface area?
Also Estimate the relative error in the calculated surface area.
To estimate the maximum error in the calculated surface area, we can use linear approximation.
The surface area of a sphere is given by the formula:
A = 4πr^2
Where A is the surface area and r is the radius.
First, we need to find the radius of the sphere. We can use the formula for the circumference of a sphere:
C = 2πr
Given that the measured circumference was 74.000 cm with a possible error of 0.50000 cm, we have:
C_actual = 74.000 cm
C_error = 0.50000 cm
Substituting this into the circumference formula, we can solve for the actual radius:
C_actual = 2πr_actual
74.000 cm = 2πr_actual
r_actual = 74.000 cm / (2π)
Now, we need to find the maximum possible radius, considering the error:
C_max = C_actual + C_error
C_min = C_actual - C_error
Substituting into the circumference formula, we can solve for the maximum and minimum radii:
C_max = 2πr_max
C_min = 2πr_min
r_max = C_max / (2π)
r_min = C_min / (2π)
With the radii obtained, we can calculate the maximum error in the surface area using linear approximation:
ΔA_max = A(r_max) - A(r_min)
ΔA_max = 4πr_max^2 - 4πr_min^2
Now, to estimate the relative error in the calculated surface area:
Relative_error = (ΔA_max / A_actual) * 100
Substitute the values and solve the equations to get the maximum error in the calculated surface area and the relative error.