Question

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

Answer 1

The percent uncertainty in the volume
`(\delta V)/(V) * 100 = 14.286\%`

Step-by-step 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 = (4\pi )/(3)* r^3`

Now taking the log of both sides

`log V = log [(4\pi)/(3) ] * r^3`

=>
`log V = log [(4\pi)/(3) ] + log( r^3)`

=>
`log V = log [(4\pi)/(3) ] + 3log( r)`

Differentiating both sides

`(1)/(V) \delta V = 0 + 3 [(1)/(r) ] \ \delta r`

Now converting them to percentage by multiplying by 100

`(\delta V)/(V) * 100 = [( 3 \delta r )/(r) ] \ * 100`

=>
`(\delta V)/(V) * 100 = [( 3 * 0.04 )/( 0.84) ] \ * 100`

=>
`(\delta V)/(V) * 100 = 14.286\%`