The circumference of a sphere was measured to

be 82.000 cm with a possible error of 0.50000 cm. Use linear
approximation to estimate the maximum error in the calculated
surface area.
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 formula for the circumference of a sphere is given by:
C = 2πr, where C is the circumference and r is the radius of the sphere.

Since the circumference was measured to be 82.000 cm with a possible error of 0.50000 cm, we can write the equation as:
82.000 cm = 2πr + ΔC, where ΔC is the possible error in the circumference measurement.

To estimate the maximum error in the calculated surface area, we need to calculate the partial derivative of the surface area with respect to the radius, and then multiply it by the maximum error in the circumference measurement.

The formula for the surface area of a sphere is given by:
A = 4πr^2, where A is the surface area and r is the radius of the sphere.

Taking the partial derivative of the surface area with respect to the radius:
∂A/∂r = 8πr.

Now, we can estimate the maximum error in the calculated surface area:
ΔA ≈ |∂A/∂r| * Δr,
where ΔA is the maximum error in the surface area, |∂A/∂r| is the magnitude of the partial derivative of the surface area with respect to the radius, and Δr is the maximum error in the circumference measurement.

Substituting the values, we get:
ΔA ≈ |8πr| * (0.50000 cm).

To estimate the relative error in the calculated surface area, we can divide the maximum error in the surface area by the actual surface area.

Relative error in surface area = ΔA / A,
where A is the actual surface area of the sphere.

To calculate the actual surface area, we can substitute the measured circumference into the formula for the surface area:
A = 4π(r^2) = (π/2)(C^2), since C = 2πr.

Now, you can substitute the values into the equations to calculate the maximum error in the calculated surface area and the relative error in the surface area.