What is the minimum volume of a block of wood(density `=850kg//m^(3))` if it is to hold a `50 kg` woman entirely above the water when she stands on it?

Explanation

To solve the problem of determining the minimum volume of a block of wood required to hold a 50 kg woman entirely above water, we can follow these steps:

### Step 1: Understand the Problem
We need to find the volume of a block of wood that can support a 50 kg woman without any part of her being submerged in water. The density of the wood is given as 850 kg/m³, and the density of water is 1000 kg/m³.

### Step 2: Apply the Principle of Buoyancy
According to Archimedes' principle, the buoyant force (upthrust) acting on the block of wood must equal the total weight that the block is supporting (the weight of the woman plus the weight of the block itself).

### Step 3: Calculate the Weight of the Woman
The weight of the woman can be calculated using the formula:
\[ \text{Weight} = \text{mass} \times g \]
Since we are working in terms of kg and kgf (kilogram-force), we can take the weight of the woman as:
\[ W_w = 50 \text{ kgf} \]

### Step 4: Calculate the Weight of the Block of Wood
The weight of the block of wood can be expressed as:
\[ W_b = V_b \times \rho_b \]
Where:
- \( V_b \) is the volume of the block of wood (which we need to find).
- \( \rho_b \) is the density of the wood, which is 850 kg/m³.

### Step 5: Calculate the Buoyant Force
The buoyant force acting on the block of wood when it is fully submerged is given by:
\[ F_b = V_b \times \rho_w \]
Where:
- \( \rho_w \) is the density of water, which is 1000 kg/m³.

### Step 6: Set Up the Equation
For the block to hold the woman entirely above water, the buoyant force must equal the total weight (weight of the woman plus weight of the block):
\[ F_b = W_w + W_b \]
Substituting the expressions we have:
\[ V_b \times \rho_w = W_w + V_b \times \rho_b \]
This simplifies to:
\[ V_b \times 1000 = 50 + V_b \times 850 \]

### Step 7: Rearranging the Equation
Rearranging the equation gives:
\[ V_b \times 1000 - V_b \times 850 = 50 \]
Factoring out \( V_b \):
\[ V_b \times (1000 - 850) = 50 \]
This simplifies to:
\[ V_b \times 150 = 50 \]

### Step 8: Solve for Volume
Now, we can solve for \( V_b \):
\[ V_b = \frac{50}{150} = \frac{1}{3} \text{ m}^3 \]
Thus, the minimum volume of the block of wood required is:
\[ V_b = 0.333 \text{ m}^3 \]

### Final Answer
The minimum volume of the block of wood required to hold a 50 kg woman entirely above water is **0.333 m³**.

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