Final answer:
The density of the olive oil is 900 kg/m³, and the density of the wooden box is 450 kg/m³, calculated using Archimedes' principle and the proportions of the box's submerged volumes in different liquids.
Step-by-step explanation:
When a wooden box floats in water with 55% of its height above the waterline and 50% above the olive oil, this situation can be explained using Archimedes' principle and the concept of density. The weight of the displaced fluid equals the weight of the object. The density of the box can be found by the ratio of the submerged volumes in water (45%) and olive oil (50%), knowing the density of water is 1000 kg/m³.
(a) The density of the olive oil (Doil) can be calculated using the following proportion: (Height of box submerged in water / Height of box submerged in oil) = (Density of olive oil / Density of water). Solving for Doil gives us Doil = (45/50) * 1000 kg/m³ = 900 kg/m³.
(b) The density of the box (Dbox) can be found through the portion submerged in water, which is 45%. This means the box has a density of 45% that of water (since it displaces an equal weight of water to its own weight when floating). Therefore, Dbox = 0.45 * 1000 kg/m³ = 450 kg/m³.