THe circumference of a sphere was measured to be 74.000 cm with a possible error of 0.50000 cm. Use the linear appoximation to estimate the maximum error in the calculated surface area. and estimate the relative error in the calculated surface area.

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delta A = 2*[(delta C)/C]*Area

= A/74

(C is the circumference; A is the area)

To estimate the maximum error in the calculated surface area, we can use the linear approximation.

The surface area of a sphere is given by the formula:
A = 4πr^2

We know the circumference of the sphere as 74.000 cm with a possible error of 0.50000 cm. Dividing the circumference by 2π will give us the radius of the sphere, as circumference = 2πr.

Let's find the radius:
r = circumference / (2π) = 74.000 cm / (2π) = 11.749 cm

Now, we need to find the maximum error in the radius by adding the possible error to the actual radius:
max_error_radius = radius_error = 0.50000 cm

Therefore, the maximum error in the radius is 0.50000 cm.

Next, we use the linear approximation formula to estimate the maximum error in the calculated surface area:
max_error_surface_area = 2A * max_error_radius

Substituting the values, we get:
max_error_surface_area = 2 * (4πr^2) * max_error_radius
max_error_surface_area = 2 * (4π * (11.749 cm)^2) * 0.50000 cm

Calculate the value to get the estimated maximum error in the surface area.

Additionally, to estimate the relative error in the calculated surface area, we can use the formula:
relative_error = max_error_surface_area / actual_surface_area

Substitute the values and calculate to get the estimated relative error in the surface area.