Answer :
Answer:
a) [tex]\%V = 87.36\,\%[/tex], b) [tex]x = 1.248\,m[/tex], c) [tex]F_{B} = 176488.341\,N[/tex], d) Six polar bears.
Explanation:
a) The slab of ice is modelled by the Archimedes' Principles and the Newton's Laws, whose equation of equilibrium is:
[tex]\Sigma F =\rho_{w}\cdot g \cdot A \cdot x-\rho_{i}\cdot g\cdot V = 0[/tex]
The height of the ice submerged is:
[tex]\rho_{w}\cdot A \cdot x = \rho_{i}\cdot V[/tex]
[tex]x = \frac{\rho_{i}\cdot V}{\rho_{w}\cdot A}[/tex]
[tex]x = \frac{\left(900\,\frac{kg}{m^{3}}\right)\cdot (20\,m^{3})}{\left(1030\,\frac{kg}{m^{3}} \right)\cdot (14\,m^{2})}[/tex]
[tex]x = 1.248\,m[/tex]
The percentage of the volume of the ice that is submerged is:
[tex]\%V = \frac{(1.248\,m)\cdot (14\,m^{2})}{20\,m^{3}} \times 100\,\%[/tex]
[tex]\%V = 87.36\,\%[/tex]
b) The height of the portion of the ice that is submerged is:
[tex]x = 1.248\,m[/tex]
c) The buoyant force acting on the ice is:
[tex]F_{B} = \left(1030\,\frac{kg}{m^{3}} \right)\cdot (1.248\,m)\cdot (14\,m^{2})\cdot \left(9.807\,\frac{m}{s^{2}} \right)[/tex]
[tex]F_{B} = 176488.341\,N[/tex]
d) The new system is modelled after the Archimedes' Principle and Newton's Laws:
[tex]\Sigma F = -n\cdot m_{bear}\cdot g-\rho_{i}\cdot V \cdot g + \rho_{w}\cdot V\cdot g = 0[/tex]
The number of polar bear is cleared in the equation:
[tex]n\cdot m_{bear} = (\rho_{w} - \rho_{i})\cdot V[/tex]
[tex]n = \frac{(\rho_{w}-\rho_{i})\cdot V}{m_{bear}}[/tex]
[tex]n = \frac{\left(1030\,\frac{kg}{m^{3}} - 900\,\frac{kg}{m^{3}} \right)\cdot (20\,m^{3})}{400\,kg}[/tex]
[tex]n = 6.5[/tex]
The maximum number of polar bears that slab could support is 6.