Question

1000 small thermocol balls of radius 0.5 cm are kept in a spherical balloon of radius 20 cm. Find the volume of air in the balloon.

Hint:

Keep the calculation in terms of \(\pi\) until the very last step.
This saves time and avoids dealing with complex decimals early in the problem.

Answer 1

Step 1: Understanding the Question:
We have a large spherical balloon containing \(1000\) tiny spherical thermocol balls.
We need to find the volume of the empty air space left inside the balloon.
The volume of air will be the total internal volume of the spherical balloon minus the combined volume of all the \(1000\) small thermocol balls.

Step 2: Key Formula or Approach:
1. The volume \(V\) of a sphere of radius \(r\) is given by:
\[ V = \frac{4}{3}\pi r^3 \] 2. Total volume of small balls:
\[ V_{\text{balls}} = 1000 \times \left(\frac{4}{3}\pi r^3\right) \] 3. Volume of the balloon:
\[ V_{\text{balloon}} = \frac{4}{3}\pi R^3 \] 4. Volume of air:
\[ V_{\text{air}} = V_{\text{balloon}} - V_{\text{balls}} \]

Step 3: Detailed Explanation:
1. Calculate the volume of one small thermocol ball of radius \(r = 0.5\text{ cm}\):
\[ v = \frac{4}{3}\pi (0.5)^3 = \frac{4}{3}\pi (0.125) = \frac{0.5}{3}\pi = \frac{1}{6}\pi\text{ cm}^3 \] 2. Calculate the total volume of \(1000\) small balls:
\[ V_{\text{balls}} = 1000 \times \frac{1}{6}\pi = \frac{500}{3}\pi\text{ cm}^3 \] 3. Calculate the volume of the large spherical balloon of radius \(R = 20\text{ cm}\):
\[ V_{\text{balloon}} = \frac{4}{3}\pi (20)^3 = \frac{4}{3}\pi (8000) = \frac{32000}{3}\pi\text{ cm}^3 \] 4. Subtract the balls' volume from the balloon's volume to find the air volume:
\[ V_{\text{air}} = V_{\text{balloon}} - V_{\text{balls}} = \frac{32000}{3}\pi - \frac{500}{3}\pi = \frac{31500}{3}\pi\text{ cm}^3 \] \[ V_{\text{air}} = 10500\pi\text{ cm}^3 \] 5. Use \(\pi = \frac{22}{7}\) to compute the final numerical value:
\[ V_{\text{air}} = 10500 \times \frac{22}{7} = 1500 \times 22 = 33000\text{ cm}^3 \] 6. Therefore, the volume of air in the balloon is \(33000\text{ cm}^3\).

Step 4: Final Answer:
The volume of air inside the balloon is \(33000\text{ cm}^3\).