Answer:
[tex]n =26786\,ballons[/tex]
Explanation:
According to the Archimedes' Principle, the buoyant force exerted on the helium ballons is equal to the weight of displaced air. A simplified scheme for the system person-ballons is presented below. The correspondent equation of equilibrium is:
[tex]\Sigma F = \rho_{Air}\cdot V_{net}\cdot g - \rho_{He}\cdot V_{net}\cdot g - m_{person}\cdot g = 0[/tex]
[tex](\rho_{Air}-\rho_{He})\cdot V_{net} - m_{person} = 0[/tex]
The minimum net volume is:
[tex]V_{net} = \frac{m_{person}}{\rho_{Air}-\rho_{He}}[/tex]
[tex]V_{net} = \frac{75\,kg}{1.2\,\frac{kg}{m^{3}}-1\,\frac{kg}{m^{3}} }[/tex]
[tex]V_{net} = 375\,m^{3}[/tex]
The volume of each ballon is:
[tex]V_{ballon} = \frac{4}{3}\pi\cdot (0.15\,m)^{3}[/tex]
[tex]V_{ballon} \approx 0.014\,m^{3}[/tex]
The minimum quantity of ballons needed to lift Josiah is:
[tex]n = \frac{V_{net}}{V_{ballon}}[/tex]
[tex]n = \frac{375\,m^{3}}{0.014\,m^{3}}[/tex]
[tex]n =26786\,ballons[/tex]
