91Ó°ÊÓ

Buoyancy is an upward force exerted by the fluid in the upward direction that opposes the weight of a fully or partially submerged body. It is equal to the weight of the water displaced. It is expressed in Newtons.

The mass of the childi s \({m_{{\rm{child}}}} = 32\;{\rm{kg}}\).

The volume of the soda bottle is \({V_{{\rm{bottle}}}} = 1.0\;{\rm{L}}\).

The buoyancy force \(\left( {{F_{{\rm{buoyant}}}}} \right)\)due to the submerged empty soda bottles must equal the weight of the child. To find the minimum number of bottles \(\left( N \right)\), we assume that each bottle is completely submerged, so displaces \(1.0\;{\rm{L}}\) of water.

\(\begin{aligned}{F_{{\rm{buoyant}}}} &= N{V_{{\rm{bottle}}}}{\rho _{{\rm{water}}}}g\\N &= \frac{{{m_{{\rm{child}}}}}}{{{V_{{\rm{bottle}}}}{\rho _{{\rm{water}}}}}}\end{aligned}\)

Substitute the values in above formula.

\(\begin{aligned}N &= \frac{{32\;{\rm{kg}}}}{{\left( {1.0\;{\rm{L}}} \right)\left( {\frac{{{{10}^{ - 3}}\;{{\rm{m}}^3}}}{{1\;{\rm{L}}}}} \right)\left( {1000\;{{{\rm{kg}}} \mathord{\left/{\vphantom {{{\rm{kg}}} {{{\rm{m}}^3}}}} \right.} {{{\rm{m}}^3}}}} \right)}}\\N &= 32\;{\rm{bottles}}\end{aligned}\)

Hence, the child will need 32 soda bottles to be able to stay dry on the raft.