To find the area of the first piston, we can use the formula:
Area = Force / (Pressure)
We are given the force exerted by the piston, which is the weight of the object it is supporting. In this case, we are given the mass of the object held by the piston (1.40 * 10^3 kg). We can calculate the weight using the equation:
Weight = Mass * Acceleration due to gravity
The acceleration due to gravity is approximately 9.8 m/s^2. Thus, the weight is:
Weight = (1.40 * 10^3 kg) * (9.8 m/s^2)
Next, we need to find the pressure exerted by the weight of the object on the piston. The pressure is equal to the weight divided by the area of the piston in contact with the object. We are told that the cross-sectional area of the ice block is the same as the piston. Therefore, the pressure can be expressed as:
Pressure = Weight / Area
We can rearrange this equation to solve for the area:
Area = Weight / Pressure
Substituting the weight and pressure values, we get:
Area = ((1.40 * 10^3 kg) * (9.8 m/s^2)) / Pressure
Now, let's find the pressure. The pressure exerted by the ice block can be calculated using the density of the ice and the thickness of the block. The pressure is given by the equation:
Pressure = Density * g * h
where density (ρ) is 917 kg/m^3 (density of the ice block), g is the acceleration due to gravity (9.8 m/s^2), and h is the thickness of the ice block, which is 0.076 m in this case.
Pressure = (917 kg/m^3) * (9.8 m/s^2) * (0.076 m)
Now, we can substitute the pressure value back into the equation to find the area:
Area = ((1.40 * 10^3 kg) * (9.8 m/s^2)) / ((917 kg/m^3) * (9.8 m/s^2) * (0.076 m))
Simplifying the equation, we find:
Area = (1.40 * 10^3 kg) / ((917 kg/m^3) * (0.076 m))
Now, calculate the area using the given values:
Area = 1.40 * 10^3 kg / (917 kg/m^3 * 0.076 m)
Area = 1.40 * 10^3 kg / 69.932 kg/m^2
Area ≈ 20.019 m^2
Therefore, the area of the first piston is approximately 20.019 m^2.