Answer:
The percent uncertainty in the volume [tex]\frac{\delta V}{V} * 100 = 14.286\%[/tex]
Explanation:
From the question we are told that
The radius is [tex]r = 0.84 \pm 0.04 \ m[/tex]
From the given value
The uncertainty of radius is [tex]\delta r = 0.04[/tex]
Generally the volume of a spherical beach ball is mathematically represented as
[tex]V = \frac{4\pi }{3}* r^3[/tex]
Now taking the log of both sides
[tex]log V = log [\frac{4\pi}{3} ] * r^3[/tex]
=> [tex]log V = log [\frac{4\pi}{3} ] + log( r^3)[/tex]
=> [tex]log V = log [\frac{4\pi}{3} ] + 3log( r)[/tex]
Differentiating both sides
[tex]\frac{1}{V} \delta V = 0 + 3 [\frac{1}{r} ] \ \delta r[/tex]
Now converting them to percentage by multiplying by 100
[tex]\frac{\delta V}{V} * 100 = [\frac{ 3 \delta r }{r} ] \ * 100[/tex]
=> [tex]\frac{\delta V}{V} * 100 = [\frac{ 3 * 0.04 }{ 0.84} ] \ * 100[/tex]
=> [tex]\frac{\delta V}{V} * 100 = 14.286\%[/tex]
