Differentials question

Question

The measurement of the radius of the end of a log is found to be 3 inches, with a possible error of 1/2 inch. Use differentials to approximate the possible propagated error in computing the area of the end of the log.

Answer 1

The end of the log is a circle with radius r. The area A of a circle is given by:

A = πr²

The differential dA represents the approximate change in the area resulting from a small change in the radius dr:

dA = (dA/dr) * dr

dA = 2πr * dr

dA = 2π * 3 * 1/2

dA = 3π

Therefore, the approximate propagated error in computing the area of the end of the log is 3π square inches.

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