To solve this problem, we will need to use some geometric relationships between the cylinder and the spheres. Let’s break down the steps:

Given:

• Four identical spherical balls are packed tightly into a cylindrical tube.
• The total surface area of the tube is $905 \, \text{cm}^2$.

Step 1: Understand the setup.

• The spheres are packed inside the cylinder in a way that they touch each other and the sides of the cylinder. This means the height of the cylinder is equal to the diameter of the four spheres, and the radius of the cylinder is equal to the radius of the spheres.

Let:

• $r$ be the radius of one of the spheres.
• $h$ be the height of the cylinder (which equals the diameter of the four spheres).

From this, we know:

• The height of the cylinder $h = 4 \times 2r = 8r$ (because there are 4 spheres stacked vertically).
• The radius of the cylinder is the same as the radius of the spheres, so the radius of the cylinder is $r$.

Step 2: Surface area of the cylinder.

The total surface area of the cylinder consists of:

1. Two circular ends (each with an area $\pi r^2$).
2. The side (lateral surface area) of the cylinder, which is $2\pi rh$.

The total surface area $A_{\text{total}}$ is: $A_{\text{total}} = 2\pi r^2 + 2\pi rh$ Substitute $h = 8r$ into this formula: $A_{\text{total}} = 2\pi r^2 + 2\pi r(8r) = 2\pi r^2 + 16\pi r^2 = 18\pi r^2$ We are given that the total surface area is $905 \, \text{cm}^2$, so: $18\pi r^2 = 905$

Step 3: Solve for $r$.

To find $r$, divide both sides by $18\pi$: $r^2 = \frac{905}{18\pi}$ $r^2 = \frac{905}{56.5487} \approx 16$ $r \approx \sqrt{16} = 4 \, \text{cm}$

Step 4: Find the volume of one ball.

The volume $V$ of a sphere is given by the formula: $V = \frac{4}{3} \pi r^3$ Substitute $r = 4 \, \text{cm}$: $V = \frac{4}{3} \pi (4)^3 = \frac{4}{3} \pi \times 64 = \frac{256}{3} \pi \approx 268.1 \, \text{cm}^3$

Final Answer:

The volume of one ball is approximately 268.1 cm³.

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Would you like a more detailed explanation of any of these steps? Or do you have another question?

Here are some related questions for further exploration:

1. How would the volume change if the number of spheres in the cylinder increased?
2. What happens to the total surface area if the radius of the spheres is halved?
3. How do you calculate the volume and surface area of a sphere from its radius?
4. How can you calculate the volume of the cylinder if only the radius and height are known?
5. What would the total surface area of the cylinder be if there were only two spheres inside?

Tip: When working with packed shapes, always remember that the arrangement affects the dimensions of the enclosing container!