Question

A spherical, helium-filled balloon has a radius of 12.0m. a mass support cables, The balloon, of and basket have 196kg. What maximum load can the balloon support while it floats at an altitude at which the helium density is 0.16kn / (m ^ 3) and the air density is 1.25kg / m ^ 3 . Assume that the volume of air displaced by the load, support cables, and basket is negligible.

Answer 1

The helium-filled balloon can support a maximum load of approximately 7241.66 kg while floating at an altitude with helium density 0.16 kg/m³ and air density 1.25 kg/m³, assuming negligible volume for the load, support cables, and basket.

1. Calculate the Volume of the Sphere:

`\[ V_{\text{sphere}} = (4)/(3) \pi r^3 \]`

Substitute
`\(r = 12.0 \ \text{m}\)` into the formula and calculate
`\(V_{\text{sphere}}\)`.

`\[ V_{\text{sphere}} = (4)/(3) \pi (12.0 \ \text{m})^3 \]\[ V_{\text{sphere}} \approx 7238.23 \ \text{m}^3 \]`

2. Calculate the Effective Density of the Displaced Air:

`\[ \rho_{\text{effective}} = \rho_{\text{air}} - \rho_{\text{helium}} \]\[ \rho_{\text{effective}} = 1.25 \ \text{kg/m}^3 - 0.16 \ \text{kg/m}^3 \]\[ \rho_{\text{effective}} = 1.09 \ \text{kg/m}^3 \]`

3. Calculate the Volume of Displaced Air:

`\[ V_{\text{displaced}} = \frac{V_{\text{sphere}}}{\rho_{\text{effective}}} \]\[ V_{\text{displaced}} = \frac{7238.23 \ \text{m}^3}{1.09 \ \text{kg/m}^3} \]\[ V_{\text{displaced}} \approx 6643.80 \ \text{m}^3 \]`

4. Calculate the Maximum Load the Balloon Can Support:

`\[ m_{\text{max load}} = \rho_{\text{effective}} \cdot V_{\text{displaced}} \]\[ m_{\text{max load}} = 1.09 \ \text{kg/m}^3 * 6643.80 \ \text{m}^3 \]\[ m_{\text{max load}} \approx 7241.66 \ \text{kg} \]`

So, the maximum load that the balloon can support while floating at the specified altitude is approximately 7241.66 kg.