91Ó°ÊÓ

The given data can be listed below,

• The volume of life preserver is,${\mathrm{V}}_{\mathrm{LP}}=0.0400{\mathrm{m}}^3$ .
• The mass of a person is, $\mathrm{m}=75\mathrm{kg}$.
• The average density of person is,${\mathrm{p}}_{\mathrm{p}}=980\mathrm{kg}/{\mathrm{m}}^3$
• The volume of a person above the water surface is 20% .

Archimedes’ principle assumes that fluids have a pressure gradient; the pressure gradient is a gradual rise in pressure from top to bottom. The pressure gradient causes buoyancy, resulting in a net up-thrust force as more pressure is exerted below than above.

This force is opposite in direction to its weight. It is a matter of density. Floating objects are those whose density exceeds water density.

The buoyant force is equal to the weight of the seawater displaced by the life preserver and 80% of the person’s volume.

Balance the forces on the person-life preserver system gives,

$\mathrm{B}={\mathrm{W}}_{\mathrm{P}}+{\mathrm{W}}_{\mathrm{lp}}$

Here, B is the buoyant force,${\mathrm{W}}_{\mathrm{p}}$ is the weight of the person and${\mathrm{W}}_{\mathrm{LP}}$ is the weight of the life preserver.

By mass-density relation,

${\mathrm{ÒÏ}}_{\mathrm{w}}({\mathrm{V}}_{\mathrm{LP}}+0.80{\mathrm{V}}_{\mathrm{P}})={\mathrm{ÒÏ}}_{\mathrm{P}}{\mathrm{V}}_{\mathrm{P}}+{\mathrm{ÒÏ}}_{\mathrm{LP}}{\mathrm{V}}_{\mathrm{LP}}$

Here,${\mathrm{ÒÏ}}_{\mathrm{W}}$ is the density of sea water,$\left(1030\mathrm{kg}/{\mathrm{m}}^3\right)$,${\mathrm{V}}_{\mathrm{LP}}$ is the volume of life preserver, ${\mathrm{V}}_{\mathrm{P}}$ is the volume of person,${\mathrm{ÒÏ}}_P$ is the density of person, and ${\mathrm{ÒÏ}}_{LP}$ is the density of life preserver.

By conversion of person’s volume above expression can be written as,

$$\begin{gathered} {\mathrm{ÒÏ}}_W=\left(V_{LP}+0.800m_p+{\mathrm{ÒÏ}}_P\right)=m_p+{\mathrm{ÒÏ}}_{LP}{V}_{LP} \\ {\mathrm{ÒÏ}}_{LP}{V}_{LP}={\mathrm{ÒÏ}}_W\left(V_{LP}+0.800m_p+{\mathrm{ÒÏ}}_P\right)-m_p \\ {\mathrm{ÒÏ}}_{LP}=\frac{{\mathrm{ÒÏ}}_W\left(V_{LP}+0.800m_p+{\mathrm{ÒÏ}}_P\right)-m_p}{V_{LP}} \end{gathered}$$

Substitute all the values in the above expression,

$$\begin{gathered} {\mathrm{ÒÏ}}_{\mathrm{LP}}=\frac{{\mathrm{ÒÏ}}_{\mathrm{w}}\left({\mathrm{V}}_{\mathrm{LP}}+0.800{\mathrm{m}}_{\mathrm{p}}/{\mathrm{ÒÏ}}_{\mathrm{p}}\right)}{{\mathrm{V}}_{\mathrm{LP}}} \\ {\mathrm{ÒÏ}}_{\mathrm{LP}}=\frac{\left(1030\mathrm{kg}/{\mathrm{m}}^3\right)\left[0.0400{\mathrm{m}}^3+{\displaystyle \frac{(0.800)(75\mathrm{kg})}{980\mathrm{kg}/{\mathrm{m}}^3}}\right]-75\mathrm{kg}}{0.400{\mathrm{m}}^3} \\ =\frac{\left(1030\mathrm{kg}/{\mathrm{m}}^3\right)\left(0.400{\mathrm{m}}^3+0.061{\mathrm{m}}^3\right)-75\mathrm{kg}}{0.400{\mathrm{m}}^3} \\ =\frac{104.26\mathrm{kg}-75\mathrm{kg}}{0.0400{\mathrm{m}}^3} \\ =732\mathrm{kg}/{\mathrm{m}}^3 \end{gathered}$$

Thus, the density of life preserver is $732 \mathrm{kg}/{\mathrm{m}}^3.$.