A supertanker filled with oil has a total mass of 6.1 × 10⁸ kg. If the dimensions of the ship are those of a rectangular box 300 meters long, 80 meters wide, and 40 meters high, determine how far the bottom of the ship is below sea level. (ρsea = 1020 kg/m³.) A)10 m B)15 m C)20 m D)25 m E)30 m

Question

Quizplus · Accepted Answer

The first step is to find the volume of the ship using its dimensions:
Volume = length × width × height
Volume = 300 m × 80 m × 40 m
Volume = 9.6 × 10^5 m^3

Next, we can use the mass and volume to determine the density of the ship:
Density = mass ÷ volume
Density = 6.1 × 10^18 kg ÷ 9.6 × 10^5 m^3
Density = 6.35 × 10^12 kg/m^3

Since the density of the ship is greater than the density of water (1020 kg/m^3), it will displace an amount of water equal to its own volume. The weight of this volume of water will push back against the ship with an equal force, resulting in a net upward force known as buoyancy.

To find the depth that the ship will sink to, we can use Archimedes' principle, which states that the buoyant force is equal to the weight of the displaced fluid:
Buoyant force = weight of displaced fluid

Weight of ship = mass of ship × gravitational acceleration
Weight of ship = 6.1 × 10^18 kg × 9.8 m/s^2
Weight of ship = 5.98 × 10^19 N

Weight of displaced fluid = density of fluid × volume of displaced fluid × gravitational acceleration
Weight of displaced fluid = 1020 kg/m^3 × 9.6 × 10^5 m^3 × 9.8 m/s^2
Weight of displaced fluid = 9.44 × 10^12 N

Buoyant force = 9.44 × 10^12 N

Since the buoyant force is equal to the weight of the displaced fluid, we can set the two expressions equal to each other and solve for the depth of the ship:
Buoyant force = Weight of ship
9.44 × 10^12 N = 5.98 × 10^19 N ÷ (300 m × 80 m)
9.44 × 10^12 N = 2.49 × 10^5 N/m
Depth = Buoyancy force ÷ (density of fluid × gravitational acceleration)
Depth = 9.44 × 10^12 N ÷ (1020 kg/m^3 × 9.8 m/s^2)
Depth ≈ 2.45 × 10^8 m/m^2
Depth ≈ 24.5 m

Therefore, the bottom of the ship is approximately 25 m below sea level. The correct answer is D) 25 m.

A Supertanker Filled with Oil Has a Total Mass of 6.1