A liquid sample has mass \(3.6\,\mathrm{kg}\) and volume \(4.5\times10^{-3}\,\mathrm{m^3}\). Find its density.

A fluid has density \(920\,\mathrm{kg\,m^{-3}}\) and occupies \(1.8\times10^{-3}\,\mathrm{m^3}\). Find its mass.

A gas sample has mass \(0.24\,\mathrm{kg}\) and density \(1.2\,\mathrm{kg\,m^{-3}}\). Find its volume.

A sample has mass \(75\,\mathrm{g}\) and volume \(50\,\mathrm{cm^3}\). Find its density in \(\mathrm{kg\,m^{-3}}\).

An empty flask has mass \(0.44\,\mathrm{kg}\). Filled with liquid, its mass is \(2.84\,\mathrm{kg}\). The flask volume is \(2.0\times10^{-3}\,\mathrm{m^3}\). Find the liquid density.

Two gas samples have the same mass. Sample A occupies \(0.30\,\mathrm{m^3}\), and sample B occupies \(0.75\,\mathrm{m^3}\). Find \(\rho_A/\rho_B\).

A rectangular tank has base area \(0.015\,\mathrm{m^2}\) and is filled to height \(0.80\,\mathrm{m}\) with brine of density \(1030\,\mathrm{kg\,m^{-3}}\). Find the mass of brine.

A spherical sealed float has radius \(4.0\,\mathrm{cm}\) and mass \(0.20\,\mathrm{kg}\). Find its average density and compare it with water density \(1000\,\mathrm{kg\,m^{-3}}\).

A gas of initial density \(1.1\,\mathrm{kg\,m^{-3}}\) is compressed from \(1.6\,\mathrm{m^3}\) to \(0.40\,\mathrm{m^3}\) without changing its mass. Find the final density.

A metal sample is lowered into a measuring cylinder. The reading rises from \(450\,\mathrm{mL}\) to \(810\,\mathrm{mL}\). If the sample mass is \(2.8\,\mathrm{kg}\), find its density.

A layer of fluid has horizontal area \(0.50\,\mathrm{m^2}\), height \(0.40\,\mathrm{m}\), and density \(\rho(y)=900+200y\), where \(y\) is in metres and \(\rho\) is in \(\mathrm{kg\,m^{-3}}\). Find the total mass.

A tank contains \(0.030\,\mathrm{m^3}\) of oil at \(780\,\mathrm{kg\,m^{-3}}\) and \(0.020\,\mathrm{m^3}\) of salt solution at \(1050\,\mathrm{kg\,m^{-3}}\). Find the average density of the combined contents.

A sealed container has outside volume \(2.5\times10^{-3}\,\mathrm{m^3}\) and total mass \(0.90\,\mathrm{kg}\). Find the average density of the sealed object and explain why it may differ from the density of the material inside.

A one-dimensional density model is \(\rho(x)=\rho_0(1+x/L)\) for \(0\le x\le L\). Derive the average density over the interval.

A cylinder of radius \(R\) and height \(H\) contains fluid with \(\rho(z)=\rho_0(1+\beta z/H)\) for \(0\le z\le H\). Derive the total mass and average density.

Two liquids of densities \(700\,\mathrm{kg\,m^{-3}}\) and \(1100\,\mathrm{kg\,m^{-3}}\) are mixed without volume change. The total volume is \(0.050\,\mathrm{m^3}\), and the average density is \(860\,\mathrm{kg\,m^{-3}}\). Find the volume of each liquid.

A gas in a vertical box of area \(A\) and height \(H\) has density \(\rho(y)=\rho_0(1-y/(2H))\). Derive the total mass, state the condition that the density stays positive on the interval, and interpret the average density.

A container of unknown empty mass \(M_c\) is filled once with volume \(V\) of liquid A and once with the same volume \(V\) of liquid B. The filled masses are \(M_A\) and \(M_B\), with densities \(\rho_A\) and \(\rho_B\). Derive \(V\) and \(M_c\) in terms of the measured quantities.

A spherical fluid cloud has density \(\rho(r)=\rho_c(1-r^2/R^2)\) for \(0\le r\le R\). Derive its total mass and average density.

A fluid parcel initially has cross-sectional area \(A\), length \(L\), and density \(\rho(x)=\rho_0(1+x/L)\). It is stretched to length \(\lambda L\) by the mapping \(x'=\lambda x\), with cross-section unchanged and mass conserved. Derive the new density \(\rho'(x')\).