Chapter 5, Problem 53E

A 200-L iron tank has a mass of 36 kg. (a) Will it float in seawater when empty? (b) When filled with freshwater? (c) When filled with gasoline?

To determine

Whether the tank will float in seawater when empty or not.

Answer to Problem 53E

The tank will float in seawater when empty.

Explanation of Solution

Given info: The volume of the tank is $200\text{L}$ and the mass of the tank is $36\text{kg}$ and the density of seawater is $1.03\times {10}^3{\text{kg/m}}^{\text{3}}$.

Write the expression for the mass of a substance in terms of volume and density.

$m=dV$

Here,

$m$ is the mass

$d$ is the density

$V$ is the volume

Substitute $1.03\times {10}^3{\text{kg/m}}^{\text{3}}$ for $d$ and $200\text{L}$ for $V$ to find the mass $m$ of $200\text{L}$ of seawater.

$\begin{array}{c}m=\left(1.03\times {10}^3{\text{kg/m}}^{\text{3}}\right)\left(200\text{L}\right)\\ =\left(1.03\times {10}^3{\text{kg/m}}^{\text{3}}\right)\left(200\text{L}\times \frac{{10}^{-3}{\text{m}}^{\text{3}}}{1\text{L}}\right)\\ =206\text{kg}\end{array}$

Thus, the empty tank can support up to $206\text{kg}$. Since its mass is only $36\text{kg}$, it will float in seawater.

Conclusion:

Therefore, the tank will float in seawater when empty.

To determine

Whether the tank will float in sea water when filled with freshwater or not.

Answer to Problem 53E

The tank will not float in seawater when filled with freshwater.

Explanation of Solution

Given info: The empty tank can support up to $206\text{kg}$ in seawater, the mass of the tank is $36\text{kg}$, and the density of freshwater is $1.00\times {10}^3{\text{kg/m}}^{\text{3}}$.

Write the expression for the mass of a substance in terms of volume and density.

$m=dV$

Substitute $1.00\times {10}^3{\text{kg/m}}^{\text{3}}$ for $d$ and $200\text{L}$ for $V$ to find the mass of $200\text{L}$ of freshwater.

$\begin{array}{c}m=\left(1.00\times {10}^{-3}{\text{kg/m}}^{\text{3}}\right)\left(200\text{L}\right)\\ =\left(1.00\times {10}^{-3}{\text{kg/m}}^{\text{3}}\right)\left(200\text{L}\times \frac{{10}^{-3}{\text{m}}^{\text{3}}}{1\text{L}}\right)\\ =200\text{kg}\end{array}$

Thus, when the tank is filled with fresh water, the total mass ($M$) of the system is,

$\begin{array}{c}M=200\text{kg}+36\text{kg}\\ =236\text{kg}\end{array}$

Since the mass tank filled with freshwater is greater than that of the maximum mass that the tank can support ($206\text{kg}$), the tank will sink in the seawater.

Conclusion:

Therefore, the tank will not float in seawater when filled with freshwater.

To determine

Whether the tank will float in sea water when filled with gasoline or not.

Answer to Problem 53E

The tank will float in seawater when filled with gasoline.

Explanation of Solution

Given info: The empty tank can support up to $206\text{kg}$ in seawater, the mass of the tank is $36\text{kg}$, and the density of gasoline is $6.8\times {10}^2{\text{kg/m}}^{\text{3}}$.

Write the expression for the mass of a substance in terms of volume and density.

$m=dV$

Substitute $6.8\times {10}^2{\text{kg/m}}^{\text{3}}$ for $d$ and $200\text{L}$ for $V$ to find the mass of $200\text{L}$ of gasoline.

$\begin{array}{c}m=\left(6.8\times {10}^2{\text{kg/m}}^{\text{3}}\right)\left(200\text{L}\right)\\ =\left(6.8\times {10}^2{\text{kg/m}}^{\text{3}}\right)\left(200\text{L}\times \frac{{10}^{-3}{\text{m}}^{\text{3}}}{1\text{L}}\right)\\ =136\text{kg}\end{array}$

Thus, when the tank is filled with fresh water, the total mass of the system is,

$\begin{array}{c}M=136\text{kg}+36\text{kg}\\ =172\text{kg}\end{array}$

Since the mass tank filled with gasoline is less than that of the maximum mass that the tank can support ($206\text{kg}$), the tank will float in the seawater.

Conclusion:

Therefore, the tank will float in seawater when filled with gasoline.