what is the percent uncertainty in the volume of a spherical beach ball whose radius is r = .84 plus or minus .04 m?

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Physics Delwyn 4 years 2021-08-06T00:25:21+00:00 2021-08-06T00:25:21+00:00 1 Answers 28 views 0

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kimchi2 · Answer 1 · 2021-08-06T00:27:15+00:00

Answer:

The percent uncertainty in the volume $\frac{\delta V}{V} * 100 = 14.286\%$

Explanation:

From the question we are told that

The radius is $r = 0.84 \pm 0.04 \ m$

From the given value

The uncertainty of radius is $\delta r = 0.04$

Generally the volume of a spherical beach ball is mathematically represented as

$V = \frac{4\pi }{3}* r^3$

Now taking the log of both sides

$log V = log [\frac{4\pi}{3} ] * r^3$

=> $log V = log [\frac{4\pi}{3} ] + log( r^3)$

=> $log V = log [\frac{4\pi}{3} ] + 3log( r)$

Differentiating both sides

$\frac{1}{V} \delta V = 0 + 3 [\frac{1}{r} ] \ \delta r$

Now converting them to percentage by multiplying by 100

$\frac{\delta V}{V} * 100 = [\frac{ 3 \delta r }{r} ] \ * 100$

=> $\frac{\delta V}{V} * 100 = [\frac{ 3 * 0.04 }{ 0.84} ] \ * 100$

=> $\frac{\delta V}{V} * 100 = 14.286\%$