An uninflated balloon of mass \(6.00 \times 10^{-3} \mathrm{~kg}\) has a volume of \(6.90 \times 10^{-2} \mathrm{~m}^{3}\) when fully inflated with a mixture of helium and hydrogen gas. You tie a long, heavy cord to the balloon and allow the balloon to float above a table. The length of cord lifted off the table is \(283 \mathrm{~mm}\), and the linear mass density of the cord is \(0.260 \mathrm{~kg} / \mathrm{m}\).

(a) What is the mass density of the gas mixture in the balloon?

(b) What is the ratio of helium to hydrogen in the mixture? The mass densities are \(1.286 \mathrm{~kg} / \mathrm{m}^{3}\) for air, \(0.179 \mathrm{~kg} / \mathrm{m}^{3}\) for helium gas, and \(0.090 \mathrm{~kg} / \mathrm{m}^{3}\) for hydrogen gas.