An 81-kg person puts on a life jacket, jumps into the water, and floats. The jacket has avolume of 3.1 x 10-2 m3 and is completely submerged under the water. The volume of theperson’s body that is under water is 6.2 x 10-2 m3. What is the density of the life jacket?
Question
An 81-kg person puts on a life jacket, jumps into the water, and floats. The jacket has avolume of 3.1 x 10-2 m3 and is completely submerged under the water. The volume of theperson’s body that is under water is 6.2 x 10-2 m3. What is the density of the life jacket?
Solution
To solve this problem, we need to use the formula for density, which is mass divided by volume.
First, we need to find the total volume of the person and the life jacket that is submerged in the water. This is given by the volume of the life jacket plus the volume of the person's body that is under water.
Total volume = volume of life jacket + volume of person's body under water Total volume = 3.1 x 10^-2 m^3 + 6.2 x 10^-2 m^3 Total volume = 9.3 x 10^-2 m^3
Next, we need to find the total mass that is submerged in the water. This is simply the mass of the person, since the life jacket is assumed to be weightless (or its weight is negligible compared to the person's weight).
Total mass = mass of person = 81 kg
Now we can find the density of the person and the life jacket combined, using the formula for density.
Density = mass/volume Density = 81 kg / 9.3 x 10^-2 m^3 Density = 870.97 kg/m^3
This is the density of the person and the life jacket combined. However, we want to find the density of the life jacket alone. To do this, we need to subtract the contribution of the person's body to the total density.
The density of the human body is typically around 985 kg/m^3. The volume of the person's body that is under water is 6.2 x 10^-2 m^3. So the contribution of the person's body to the total density is:
Density contribution of person's body = density of body x volume of body under water Density contribution of person's body = 985 kg/m^3 x 6.2 x 10^-2 m^3 Density contribution of person's body = 61.07 kg
Subtracting this from the total mass gives us the mass of the life jacket:
Mass of life jacket = total mass - mass contribution of person's body Mass of life jacket = 81 kg - 61.07 kg Mass of life jacket = 19.93 kg
Finally, we can find the density of the life jacket by dividing its mass by its volume:
Density of life jacket = mass of life jacket / volume of life jacket Density of life jacket = 19.93 kg / 3.1 x 10^-2 m^3 Density of life jacket = 642.58 kg/m^3
So the density of the life jacket is approximately 642.58 kg/m^3.
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