A team of 6 scouts plans to cross a lake on a raft they designed. The scouts have wooden beams with an average relative density of 0.80. The beams measure 30cmx30cmx3m. The average scouts' mass is about 65kg and for their safety, they want the top of the raft to be at least 3cm above the surface of the water. How many beams do they need to complete their projects successfully?
To determine how many beams the scouts need to complete their raft project successfully, we need to calculate the total weight of the scouts and compare it to the buoyant force exerted by the water on the raft.
Let's break down the steps to find the answer:
1. Calculate the volume of a single wooden beam:
The volume (V) of a rectangular prism is calculated by multiplying the length (L) by the width (W) by the height (H). In this case, the length is 3 meters, the width is 30 centimeters (or 0.3 meters), and the height is 30 centimeters (or 0.3 meters).
Hence, V = L × W × H = 3m × 0.3m × 0.3m = 0.27 cubic meters.
2. Calculate the mass of a single wooden beam:
The mass (m) of a wooden beam can be calculated by multiplying its volume (V) by its relative density (ρ). In this case, the relative density is given as 0.80 and the volume is 0.27 cubic meters.
Thus, m = ρ × V = 0.80 × 0.27 cubic meters = 0.216 cubic meters.
3. Determine the total weight of the scouts:
Since there are 6 scouts and their average mass is 65kg, the total weight (W) can be calculated by multiplying the average mass by the number of scouts.
Therefore, W = 6 scouts × 65kg = 390 kg.
4. Calculate the buoyant force exerted by the water:
The buoyant force (F_b) exerted by the water is equal to the weight of the water displaced by the raft when it floats. The weight of the water displaced is equal to the weight of the raft.
Hence, F_b = W = 390 kg.
5. Determine the total volume of the raft:
Since the raft is made up of wooden beams, we need to find the total volume of all the beams combined. This can be calculated by multiplying the volume of a single beam by the number of beams required.
Let's assume the number of beams required is X.
Therefore, total volume = X beams × 0.27 cubic meters/beam.
6. Calculate the required number of beams:
To find the number of beams needed, we need to equate the weight of the scouts to the buoyant force exerted by the water.
We can set up an equation as follows: W = X beams × 0.216 kg/beam.
Substituting the known values, we have 390 kg = X beams × 0.216 kg/beam.
Solving for X beams:
Dividing both sides of the equation by 0.216 kg/beam, we get:
X beams = 390 kg / 0.216 kg/beam ≈ 1805.55 beams.
Since we cannot have a fraction of a beam, the team will need at least 1806 beams to complete their raft project successfully.
Therefore, the scouts need at least 1806 beams to ensure the top of the raft is at least 3cm above the surface of the water and can support their weight safely.