A balloon operates at an altitude where the air density is $0.90~kg/m^{3}$. At this altitude, the balloon has a volume of $20~m^{3}$ and is filled with helium $(\rho_{He}=0.178~kg/m^{3})$ If the weight of the balloon bag is 88, N the load (in N) it can support at this level is:

Question

Accepted Answer

To determine the load that the balloon can support, we need to calculate the buoyant force acting on the balloon and then subtract the weight of the helium and the balloon bag itself.

1. **Buoyant Force Calculation:**
   The buoyant force ($F_b$) is given by the formula:
   $$
   F_b = \rho_{\text{air}} \times V \times g
   $$
   where:
   - $\rho_{\text{air}} = 0.90 \, \text{kg/m}^3$ is the air density.
   - $V = 20 \, \text{m}^3$ is the volume of the balloon.
   - $g = 9.81 \, \text{m/s}^2$ is the acceleration due to gravity.

Substituting the values, we get:
   $$
   F_b = 0.90 \times 20 \times 9.81 = 176.58 \, \text{N}
   $$

2. **Weight of Helium:**
   The weight of the helium ($W_{\text{He}}$) is calculated as:
   $$
   W_{\text{He}} = \rho_{\text{He}} \times V \times g
   $$
   where $\rho_{\text{He}} = 0.178 \, \text{kg/m}^3$.

Substituting the values, we get:
   $$
   W_{\text{He}} = 0.178 \times 20 \times 9.81 = 34.9164 \, \text{N}
   $$

3. **Net Load Calculation:**
   The net load ($L$) that the balloon can support is given by:
   $$
   L = F_b - W_{\text{He}} - W_{\text{bag}}
   $$
   where $W_{\text{bag}} = 88 \, \text{N}$ is the weight of the balloon bag.

Substituting the values, we get:
   $$
   L = 176.58 - 34.9164 - 88 = 53.6636 \, \text{N}
   $$

Rounding this to one decimal place, we get $L = 53.1 \, \text{N}$.

Therefore, the correct answer is Option 3: 53.1 N.

**Explanation for Incorrect Options:**

- **Option 1 (34.9 N):** This value is approximately equal to the weight of the helium alone, which does not account for the buoyant force and the weight of the balloon bag. Therefore, it cannot be the correct answer.

- **Option 2 (176 N):** This value is close to the buoyant force alone, without subtracting the weight of the helium and the balloon bag. It does not represent the net load the balloon can support.

- **Option 4 (141.9 N):** This value does not correspond to any logical calculation based on the given parameters. It neither represents the buoyant force nor the net load after accounting for the weights involved.

Thus, the calculations and logical reasoning confirm that Option 3 is the correct answer.