Question

Find th approximate volume of metal in a hollow spherical shell whose internal and external radii are $3\text{cm}$ and $3.0005\text{cm}$,respectively.

Answer 1

Volume of spherical shell
$\text{V}=\frac{4}{3}\pi [{\text{r}}_2^3-{\text{r}}_1^3]=\frac{4}{3}\pi \left[\right(3.0005{)}^3-\left(3{)}^3\right]$...(i)
Now, finding approximate value of $\left(3.0005\right)$.
Let $\text{y}={\text{x}}^3$
where, $\text{x}=3$ & $\Delta \text{x}=0.0005$
We know,
$\Delta \text{y}=\text{(x}+\Delta \text{x}{)}^3-(\text{x}{)}^3$
$\Rightarrow \Delta \text{y}=(3.0005{)}^3-(3{)}^3$
$\Rightarrow \Delta \text{y}=(3.0005{)}^3-27$...(ii)
Now, $\Delta \text{y}\approx \left(\frac{\text{dy}}{\text{dx}}\right)\Delta \text{x}$
$\Rightarrow \Delta \text{y}\approx \left(\frac{{\text{d(x}}^3)}{\text{dx}}\right)\left(0.0005\right)$
$\Rightarrow \Delta \text{y}\approx {\text{(3x}}^2\left)\right(0.0005)$
$\Rightarrow \Delta \text{y}\approx {\text{(3(3)}}^2\left)\right(0.0005)$
$\Rightarrow \Delta \text{y}\approx \left(27\right)\left(0.0005\right)$
$\Delta \text{y}\approx \left(27\right)\left(0.0005\right)$
Substituting value of $\Delta \text{y}$ & $\Delta \text{x}$ in equtaion(ii)
$\Rightarrow 0.0135\approx (3.0005{)}^3-27$
$27+0.0135\approx (3.0005{)}^3$
$\Rightarrow (3.0005{)}^3\approx 27.0135$
Substituting approximate value of $(3.0005{)}^3$ in equation (i).
$\text{V}\approx \frac{4}{3}\pi [27.0135-27.000]$
$\text{V}\approx \frac{4}{3}\pi \left[0.0135\right]$
$\text{V}\approx 4\pi \times 0.0045$
$\text{V}\approx 0.0180\pi {\text{cm}}^3$

Hence, the approximate volume of hollow spherical shell is $0.0180\pi {\text{cm}}^3$