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The question in the image asks to calculate the maximum volume of styrofoam needed to fill a bean bag chair given that the chair has a volume of 195 liters and is filled with tiny styrofoam spheres, each having a diameter of 1.2 mm. It mentions that the maximum fraction of volume that can be filled with spheres is 74%.

To calculate the volume of styrofoam needed, we can apply the following steps:

### Steps:
1. The maximum fraction of the chair's volume that can be filled by the spheres is 74%. This means 26% of the volume will be air (unfilled space).
   
2. The total volume of the bean bag chair is 195 liters, and only 74% of this volume will be filled by the spheres.

3. To find the volume of styrofoam, we multiply the total volume of the chair by 0.74 (representing 74%).

### Calculation:
\[
\text{Volume of styrofoam} = 195 \, \text{L} \times 0.74 = 144.3 \, \text{L}
\]

Thus, the maximum volume of styrofoam needed is **144 liters** to three significant figures.

Would you like more details or have any questions?

Here are 5 related questions to expand on this:
1. How does the packing fraction change with different sphere sizes?
2. What would happen to the volume if the spheres were arranged differently, like in a cubic packing?
3. What assumptions are made in calculating the packing fraction for spheres?
4. How would you calculate the total surface area of the styrofoam spheres?
5. How does the diameter of spheres affect the number of spheres needed to fill the chair?

**Tip:** When solving packing problems involving spheres, always remember to account for the unfilled spaces between them!

A bean bag chair is filled with tiny styrofoam spheres 1.2 mm in diameter. All of the spheres have the same diameter. If the bean bag chair has a volume of 195 L, what is the maximum volume of styrofoam that would be needed to fill this chair? Express your final answer in L to three significant figures and do not include units in your final answer.

Learn how to calculate the maximum volume of styrofoam needed to fill a 195 L bean bag chair with tiny 1.2 mm diameter spheres. This solution covers key concepts like volume calculation, packing fraction (74%), and step-by-step calculations. Suitable for students in Grades 9-11.

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