If two different objects have different mass distributions described by their respective density functions, would a dozen of each object necessarily have the same total mass?

• A. No, because the total mass depends on the integral of the density function over the object's volume, which can differ for each object. Specifically, the total mass is given by $\int {\rho }_{\mathrm{object}}\mathrm{dV}$.
• B. No, but only if the objects are made of different materials, not because of their mass distribution.
• C. Yes, as long as the objects have the same shape, their total mass will be equal for a dozen of each.
• D. Yes, a dozen of any object will always have the same mass regardless of their mass distribution.

Answer: (A)

Verified step by step guidance

1. Understand that the total mass of an object is determined by integrating its density function over its entire volume. Mathematically, this is expressed as \(M = \int \rho_{object} \, dV\), where \(\rho_{object}\) is the density function and \(dV\) is a volume element.
2. Recognize that if two objects have different density functions, their mass distributions vary throughout their volumes, which means the integral of their densities over their volumes can yield different total masses.
3. Note that having the same number of objects (a dozen of each) simply multiplies the total mass of one object by twelve, so the total mass for each set depends directly on the individual object's mass.
4. Conclude that because the density functions differ, the total mass of a dozen of one object will generally not be equal to the total mass of a dozen of the other object, even if the number of objects is the same.
5. Remember that the material composition and density distribution are key factors in determining mass, not just the count or shape of the objects.