To solve the problem step by step, we will calculate the required quantities one by one.
### Step 1: Calculate the mass of hydrogen in the balloon.
The mass of hydrogen can be calculated using the formula:
\[
\text{Mass of hydrogen} (m_h) = \text{Density of hydrogen} \times \text{Volume of balloon}
\]
Given:
- Density of hydrogen = \(0.09 \, \text{kg/m}^3\)
- Volume of balloon = \(15 \, \text{m}^3\)
Substituting the values:
\[
m_h = 0.09 \, \text{kg/m}^3 \times 15 \, \text{m}^3 = 1.35 \, \text{kg}
\]
### Step 2: Calculate the mass of hydrogen and the balloon.
The total mass of hydrogen and the balloon can be calculated as:
\[
\text{Total mass} = \text{Mass of hydrogen} + \text{Mass of balloon}
\]
Given:
- Mass of empty balloon = \(7.15 \, \text{kg}\)
Substituting the values:
\[
\text{Total mass} = 1.35 \, \text{kg} + 7.15 \, \text{kg} = 8.50 \, \text{kg}
\]
### Step 3: Calculate the total mass of hydrogen, balloon, and equipment.
Let the mass of the equipment be \(x \, \text{kg}\). The total mass can be expressed as:
\[
\text{Total mass} = \text{Mass of hydrogen} + \text{Mass of balloon} + \text{Mass of equipment}
\]
Substituting the known values:
\[
\text{Total mass} = 1.35 \, \text{kg} + 7.15 \, \text{kg} + x = 8.50 \, \text{kg} + x
\]
### Step 4: Calculate the mass of air displaced by the balloon.
The mass of air displaced can be calculated using the formula:
\[
\text{Mass of air displaced} = \text{Density of air} \times \text{Volume of balloon}
\]
Given:
- Density of air = \(1.3 \, \text{kg/m}^3\)
Substituting the values:
\[
\text{Mass of air displaced} = 1.3 \, \text{kg/m}^3 \times 15 \, \text{m}^3 = 19.5 \, \text{kg}
\]
### Step 5: Calculate the mass of the equipment using the law of flotation.
According to the law of flotation, the total weight of the balloon (hydrogen + balloon + equipment) is equal to the weight of the air displaced. Thus:
\[
\text{Mass of air displaced} = \text{Mass of hydrogen} + \text{Mass of balloon} + \text{Mass of equipment}
\]
Substituting the known values:
\[
19.5 \, \text{kg} = 1.35 \, \text{kg} + 7.15 \, \text{kg} + x
\]
\[
19.5 \, \text{kg} = 8.50 \, \text{kg} + x
\]
Solving for \(x\):
\[
x = 19.5 \, \text{kg} - 8.50 \, \text{kg} = 11 \, \text{kg}
\]
### Summary of Results:
1. Mass of hydrogen = \(1.35 \, \text{kg}\)
2. Mass of hydrogen and balloon = \(8.50 \, \text{kg}\)
3. Total mass of hydrogen, balloon, and equipment = \(19.5 \, \text{kg}\)
4. Mass of air displaced by balloon = \(19.5 \, \text{kg}\)
5. Mass of equipment = \(11 \, \text{kg}\)
