Question

A 32-kg child decides to make a raft out of empty 1.0-L soda bottles and duct tape. Neglecting the mass of the duct tape and plastic in the bottles, what minimum number of soda bottles will the child need to be able stay dry on the raft?

Short answer

The minimum number of soda bottles that the child needis $32$.

Step-by-step solution

Step1:Understanding Newton's Second Law

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.

Step2:Identification of the given data

The mass of the childi s $m_{\mathrm{c}\mathrm{h}\mathrm{i}\mathrm{l}\mathrm{d}}=32\mathrm{k}\mathrm{g}$.

The volume of the soda bottle is $V_{\mathrm{b}\mathrm{o}\mathrm{t}\mathrm{t}\mathrm{l}\mathrm{e}}=1.0\mathrm{L}$.

Determination of the number of soda bottles

The buoyancy force $\left(F_{\mathrm{b}\mathrm{u}\mathrm{o}\mathrm{y}\mathrm{a}\mathrm{n}\mathrm{t}}\right)$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 $1.0\mathrm{L}$ of water.

$\begin{array}{rl}{F}_{\mathrm{b}\mathrm{u}\mathrm{o}\mathrm{y}\mathrm{a}\mathrm{n}\mathrm{t}}& =N{V}_{\mathrm{b}\mathrm{o}\mathrm{t}\mathrm{t}\mathrm{l}\mathrm{e}}{\rho }_{\mathrm{w}\mathrm{a}\mathrm{t}\mathrm{e}\mathrm{r}}g\\ N& =\frac{m_{\mathrm{c}\mathrm{h}\mathrm{i}\mathrm{l}\mathrm{d}}}{V_{\mathrm{b}\mathrm{o}\mathrm{t}\mathrm{t}\mathrm{l}\mathrm{e}}{\rho }_{\mathrm{w}\mathrm{a}\mathrm{t}\mathrm{e}\mathrm{r}}}\end{array}$

Substitute the values in above formula.

$\begin{array}{rl}N& =\frac{32\mathrm{k}\mathrm{g}}{\left(1.0\mathrm{L}\right)\left(\frac{{10}^{-3}{\mathrm{m}}^3}{1\mathrm{L}}\right)\left(1000\mathrm{k}\mathrm{g}/\phantom{\mathrm{k}\mathrm{g}{\mathrm{m}}^3}{\mathrm{m}}^3\right)}\\ N& =32\mathrm{b}\mathrm{o}\mathrm{t}\mathrm{t}\mathrm{l}\mathrm{e}\mathrm{s}\end{array}$

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