Answer:
The boat's bottom is approximately 0.664 meters (or 66.4 centimeters) below the surface of the water when it's floating.
Step-by-step explanation:
To determine how far below the surface of the water the boat's bottom is when it's floating, we can use Archimedes' principle, which states that the buoyant force on an object immersed in a fluid (like water) is equal to the weight of the fluid displaced by the object.
Let:
W_boat = Weight of the boat and its cargo (in Newtons, N) = 5200 N
A_bottom = Area of the boat's bottom (in square meters, m²) = 8 m²
ρ_water = Density of water (in kilograms per cubic meter, kg/m³) ≈ 1000 kg/m³ (approximately)
The buoyant force (B) acting on the boat is given by:
B = ρ_water * V_displaced * g
Where:
ρ_water = Density of water
V_displaced = Volume of water displaced by the boat (when it's floating)
g = Acceleration due to gravity ≈ 9.81 m/s²
The weight of the boat and its cargo is balanced by the buoyant force, so we have:
W_boat = B
Now, we can find V_displaced using the equation for the volume of water displaced:
V_displaced = A_bottom * x
Substituting this into the equation for the buoyant force, we get:
W_boat = ρ_water * (A_bottom * x) * g
Now we can solve for x:
x = W_boat / (ρ_water * A_bottom * g)
Let's plug in the values:
x = 5200 N / (1000 kg/m³ * 8 m² * 9.81 m/s²)
Now, let's calculate x:
x ≈ 0.664 meters
So, the boat's bottom is approximately 0.664 meters (or 66.4 centimeters) below the surface of the water when it's floating.