5. The circumference of a sphere was measured to be 84 cm with a possible error of 0.5 cm. (a) Use differentials to estimate the maximum error in the calculated surface area. What is the relative error

(a) The maximum error in the calculated surface area is 168 square centimeters, (b) The relative error is 7.48 per cent.

Step-by-step explanation:

(a) The surface area of a sphere (), measured in square centimeters, is represented by the following formula:

Where is the radius of the sphere, measured in centimeters.

The differential is obtained by this:

Where is the partial derivative of the surface area of the sphere in terms of the radius.

The circumference () of a sphere, measured in centimeters, is represented by this expression:

Where is the radius of the sphere, measured in centimeters, and which is now cleared:

If , the radius of the sphere is:

Given that and , the surface area of the sphere is:

The maximum error in the calculated surface area is 168 square centimeters.

(b) The relative error, measured in percentage, is given by this equation:

The surface area of the sphere is: ()

If and , then:

The relative error is 7.48 per cent.

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