Question

The measurement of the radius of the end of a log is found to be 9 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

Answer:

$\pm 9in^2$

Step-by-step explanation:

We are given that

Radius of end of a log, r= 9 in

Error, $\Delta r=\pm 1/2$in

We have to find the error in computing the area of the end of the log by using differential.

Area of end of the log, A=$pi r^2$

$\frac{dA}{dr}=2\pi r$

$\frac{dA}{dr}=2\pi (9)=18\pi in^2$

Now,

Approximate error in area

$dA=\frac{dA}{dr}(\Delta r)$

Using the values

$dA=18\pi (\pm 1/2)$

$\Delta A\approx dA=\pm 9in^2$

Hence, the possible propagated error in computing the area of the end of the log$=\pm 9in^2$

Answer 2

Answer:

$A = (254.34 \pm 28.26) in^2$

Step-by-step explanation:

radius, r = 9 inches

error = 0.5 inch

The area of the end is

A = 3.14 x r x r = 3.14 x 9 x 9 = 254.34 in^2

$A = \pi r^2\\\\\frac{dA}{dr}=2\pi r\\dA = 2 pi r dr \\\\dA = 2 \times 3.14\times 9\times 0.5 = 28.26$

So, the area is given by

$A = (254.34 \pm 28.26) in^2$