A flat slab of styrofoam, with a density of 32 kg/m^3, floats on a lake. What is the minimum volume the slab must have so that a 42 kg boy can sit on the slab without it sinking?
bouyancy-42g=0
density water*g*volume-32g*volume-42g=0
volume= 42/(densitywater-32kg/m^3)
density water = 1000kg/m^3 about.
i have the same question, and i got .043 m^3. but it's not the right answer. what did i do wrong? ( i use the same formula too.)
nvm. i submit the answer wrong.
To determine the minimum volume the slab must have to support the weight of the boy without sinking, we need to consider the concept of buoyancy.
Buoyancy is the upward force exerted on an object immersed in a fluid (in this case, the lake). According to Archimedes' principle, this buoyant force is equal to the weight of the fluid displaced by the object. If the buoyant force is greater than or equal to the weight of the object, it will float; otherwise, it will sink.
In this scenario, the buoyant force should be at least equal to the weight of both the slab and the boy to keep them afloat. First, let's calculate the weight of the slab using its density and volume:
Weight of slab = density × volume × g,
where density = 32 kg/m³ and g = acceleration due to gravity (9.8 m/s²).
Since we want to find the minimum volume, let's assume the boy and the slab are a solid object, and the effective density is the combined density of both. We can represent this density as:
Effective density = (density of boy × weight of boy + density of slab × weight of slab) / (weight of boy + weight of slab).
We want the effective density to match the density of the lake, which is 1000 kg/m³ since the slab should just float.
Applying this condition, we have:
1000 kg/m³ = (density of boy × 42 kg + 32 kg/m³ × weight of slab) / (42 kg + weight of slab).
Now, we can rearrange the equation to solve for the weight of the slab:
32 kg/m³ × weight of slab = (1000 kg/m³ - density of boy × 42 kg) × (42 kg + weight of slab).
Simplifying further:
32 kg/m³ × weight of slab = (42000 kg - density of boy × 42 kg) + weight of slab × (1000 kg/m³ - density of boy × 42 kg).
Let's say the density of the boy is "d" kg/m³. Substituting this into the equation:
32 kg/m³ × weight of slab = (42000 kg - d × 42 kg) + weight of slab × (1000 kg/m³ - d × 42 kg).
Now, we can solve this equation for the weight of the slab, which will give us the minimum volume:
weight of slab = [42000 kg - d × 42 kg] / [1 - (32 kg/m³ - d × 42 kg)].
Using the given values, let's say the density of the boy is approximately the density of water, which is 1000 kg/m³. Substituting this value:
weight of slab = [42000 kg - 1000 kg/m³ × 42 kg] / [1 - (32 kg/m³ - 1000 kg/m³ × 42 kg)].
Simplifying further:
weight of slab = [42000 kg - 42000 kg] / [1 - (32 kg/m³ - 42000 kg/m³)].
weight of slab = 0 kg / [1 - (-41968 kg/m³)].
The weight of the slab is 0 kg, meaning it doesn't need any volume to support the weight of the boy without sinking. This implies that even a minimal volume slab made of styrofoam will be sufficient to keep the boy afloat in this scenario.