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Homework Statement
A submarine is in water, depth 30 meters. Inside submarine there is default air pressure. Submarine has volume of 125 m³, from which 10 m³ is water tank used for submerging. Submarine weights 123 tons. How large portion of the water tank has to be filled with sea water in order to stay still?
Homework Equations
Density:
$$ \rho = \frac{m}{v} $$
Archimede's principle
The Attempt at a Solution
By comparing densities we can determine if a object in fluid is sinking, floating or staying still. If
$$ \rho_{\text{obj}} > \rho_{\text{fluid}} \implies \text{sinking} $$
$$ \rho_{\text{obj}} < \rho_{\text{fluid}} \implies \text{floating} $$
$$ \rho_{\text{obj}} = \rho_{\text{fluid}} \implies \text{staying still} $$
. This can be derived from Archimede's principle.
Now we want density of the submarine to be equal to density of water. Which means it stays still in relation to the water. In following equation left hand side (density of submarine), right hand side (density of water). M = mass of submarine, m = mass of water in the tank, v = volume of the submarine.
$$ \frac{M+m}{v} = \rho_{\text{v}} $$
Mass of the water can be expressed as $$m = \rho_{\text{fluid}}v_\text{water in tank}$$
$$ \frac{M+\rho_{\text{fluid}}v_\text{water in tank}}{v} $$
$$ \implies v_\text{water in tank} = v-\frac{M}{\rho_{\text{fluid}}} $$
By computing this we will have
$$ v_\text{water in tank} \approx 1.63 \text{ m}^3 $$, which is 16.3% from the 10 m³ tank. Correct answer would be 56% of the 10m³ tank. Can't exactly see what I'am doing wrong?