91Ó°ÊÓ

When it comes to floating objects, buoyancy is a key concept. Buoyancy is the upward force exerted on an object immersed in a fluid, which, in our case, is the water in the lake. This force is crucial because it determines whether an object will float or sink. The principle that explains this force is Archimedes' principle.
According to Archimedes' principle, the buoyant force on an object in a fluid is equal to the weight of the fluid displaced by that object. This means if the upward buoyant force is equal to or greater than the downward force of gravity on the object, the object will float.
In the context of our problem, the slab of ice and the woman together must displace enough water to balance their combined weights. To achieve this, the volume of water displaced must be sufficient to produce a buoyant force equal to the weight of both the ice and the woman.

Density plays a significant role in determining buoyancy. Density is defined as mass per unit volume and is often represented by the symbol \( \rho \). For ice, the density \( \rho_{ice} \) is approximately 917 kg/m³. This is crucial for understanding how ice floats.
Since ice is less dense than water (which has a density of about 1000 kg/m³), it is able to float. This difference in density means that ice displaces a lesser volume of water when compared to an equal mass of liquid water.
In our problem, we need to ensure that the ice slab, with the woman standing on it, doesn't sink. We use the density of ice to calculate how much ice is needed to displace enough water to equal the weight of the ice plus the woman. This involves using the equation: \( F_{ice} = \rho_{ice} \times V_{ice} \times g \), where \( V_{ice} \) is the volume of the ice slab.

The weight of displaced water is a central factor in applying Archimedes' principle. This weight is essentially what provides the upward buoyant force.
To support a given weight on the water, the amount of water displaced must have a weight equal to that of the object being supported. This means that to keep the woman dry, the ice needs to displace enough water almost equal to or greater than the weight of both the ice and the woman.
In mathematical terms, the weight of the displaced water is given by \( F_{displaced\,water} = (\rho_{water} \times V_{ice}) \times g \). Here \( \rho_{water} = 1000 \text{ kg/m}^3 \) is the density of water, and \( V_{ice} \) is the volume of ice we are calculating. Solving the balance of forces, \( F_{woman} + F_{ice} = F_{displaced\,water} \), allows us to find the minimum volume of the ice needed to achieve this equilibrium.