Understanding the Problem

The question asks us to determine how many small hollow spheres can be completely filled with water from a single large hollow sphere that is completely filled. This involves comparing the volumes of the spheres. The total volume of water available is the volume of the large sphere. This volume will be distributed among the smaller spheres.

Sphere Volume Calculation

The volume of a sphere is given by the formula:

\( V = \frac{4}{3}\pi r^3 \)

where \( V \) is the volume and \( r \) is the radius of the sphere.

Identifying Sphere Dimensions

• The small hollow spheres have an inner radius of 1 cm. Let \( r_s \) be the radius of a small sphere. So, \( r_s = 1 \text{ cm} \).
• The large hollow sphere has an inner diameter of 20 cm. Let \( D_L \) be the diameter of the large sphere. So, \( D_L = 20 \text{ cm} \).

The radius of the large sphere, \( R_L \), is half of its diameter:

\( R_L = \frac{D_L}{2} = \frac{20 \text{ cm}}{2} = 10 \text{ cm} \)

Calculating Sphere Volumes

Let's calculate the volume of one small sphere, \( V_s \), using its radius \( r_s = 1 \text{ cm} \):

\( V_s = \frac{4}{3}\pi (r_s)^3 = \frac{4}{3}\pi (1)^3 = \frac{4}{3}\pi \times 1 = \frac{4}{3}\pi \text{ cm}^3 \)

Now, let's calculate the volume of the large sphere, \( V_L \), using its radius \( R_L = 10 \text{ cm} \):

\( V_L = \frac{4}{3}\pi (R_L)^3 = \frac{4}{3}\pi (10)^3 = \frac{4}{3}\pi \times 1000 \text{ cm}^3 \)

Determining the Number of Spheres

The total volume of water from the large sphere is \( V_L \). If this water is used to fill \( N \) small spheres, the total volume of water in these small spheres will be \( N \times V_s \). Since all the water from the large sphere is used to fill the small spheres, we have:

\( V_L = N \times V_s \)

Substitute the calculated volumes:

\( \frac{4}{3}\pi \times 1000 = N \times \frac{4}{3}\pi \)

To find \( N \), we can divide both sides of the equation by \( \frac{4}{3}\pi \):

\( N = \frac{\frac{4}{3}\pi \times 1000}{\frac{4}{3}\pi} \)

\( N = 1000 \)

Therefore, 1000 hollow spheres having an inner radius of 1 cm can be completely filled by transferring water from a completely filled hollow sphere having an inner diameter of 20 cm.