# A 2.8 kg air mattress is 2 m long, .5 m wide, & .1 m thick. What mass can it support in water before sinking?

## 10000

## To calculate the mass that an air mattress can support in water before sinking, we need to consider the buoyant force acting on the mattress. The buoyant force is equal to the weight of the water displaced by the mattress.

Step 1: Calculate the volume of the air mattress.

The volume of the air mattress can be calculated by multiplying its length, width, and thickness.

Volume = Length x Width x Thickness

Volume = 2m x 0.5m x 0.1m

Volume = 0.1 cubic meters

Step 2: Calculate the weight of the water displaced by the air mattress.

The weight of the water displaced is equal to the buoyant force acting on the air mattress.

Weight of water = Volume x Density of water x Acceleration due to gravity

The density of water is approximately 1000 kg/m^3, and the acceleration due to gravity is approximately 9.8 m/s^2.

Weight of water = 0.1m^3 x 1000kg/m^3 x 9.8 m/s^2

Weight of water = 98 N

Step 3: Convert the weight of the water to mass.

The weight of an object is equal to its mass multiplied by the acceleration due to gravity.

Weight = Mass x Acceleration due to gravity

Mass = Weight / Acceleration due to gravity

Mass = 98 N / 9.8 m/s^2

Mass = 10 kg

Therefore, the air mattress can support a mass of 10 kg in water before sinking.

## To find out the mass the air mattress can support in water before sinking, we need to determine the net buoyant force acting on it. The net buoyant force is the difference between the weight of the air mattress and the weight of the water it displaces.

Here's how we can calculate it step by step:

1. Calculate the volume of the air mattress:

The volume (V) of a rectangular prism can be found by multiplying its length (L), width (W), and thickness (T):

V = L * W * T

In this case, the length (L) is 2 m, the width (W) is 0.5 m, and the thickness (T) is 0.1 m.

Substituting these values, we have:

V = 2 * 0.5 * 0.1

Thus, the volume of the air mattress is 0.1 cubic meters (m^3).

2. Calculate the weight of the water displaced by the air mattress:

The weight of an object can be calculated using the formula:

Weight = mass * gravity

Since the density (ρ) of water is approximately 1000 kg/m^3 and the volume (V) of the air mattress is 0.1 m^3, the mass (m_displaced) of the water displaced by the air mattress is:

m_displaced = ρ * V

Therefore, the mass of the water displaced is 100 kg (1000 kg/m^3 * 0.1 m^3).

And the weight of the displaced water is:

Weight_displaced = m_displaced * gravity

3. Calculate the weight of the air mattress:

The weight of the air mattress can be calculated using its mass (m_air_mattress) and gravity. It is given in the problem as 2.8 kg.

Weight_air_mattress = m_air_mattress * gravity

4. Calculate the net buoyant force:

The net buoyant force (F_net) acting on the air mattress can be found by taking the difference between the weight of the water displaced and the weight of the air mattress:

F_net = Weight_displaced - Weight_air_mattress

5. Determine the maximum mass the air mattress can support without sinking:

Since the net buoyant force should be greater than or equal to zero for the air mattress to float, the maximum mass it can support can be calculated by dividing the net buoyant force by gravity:

Maximum_mass = F_net / gravity

Now you can use these steps to plug in the values and calculate the maximum mass the air mattress can support in water before sinking.

## It can support a mass M such that

M + m = volume x (water density)

where m is the mass of the mattress and Volume = 0.1 m^3. The water density is 1000 kg/m^2.

Solve for M