Did you know?
The cost of painting the surface area of a sphere is directly proportional to its surface area. In this case, the cost of painting the surface area of a small sphere is 9/16 of the cost of painting the surface area of a large sphere.
To find the diameter of the small sphere, we can use the proportional relationship between the surface area and the diameter. Since the surface area of the large sphere is proportional to the square of its diameter, we can set up the following equation:
(Small sphere surface area) / (Large sphere surface area) = (Small sphere diameter)² / (Large sphere diameter)²
Simplifying this equation using the given information, we have:
(9/16) = (Small sphere diameter)² / (24 cm)²
Solving for the diameter of the small sphere, we find that it is equal to 18 cm.
Next, to find the volume of the small sphere, we can use the proportional relationship between the volume and the cube of the diameter. Since the volume of the large sphere is proportional to the cube of its diameter, we can set up the following equation:
(Small sphere volume) / (Large sphere volume) = (Small sphere diameter)³ / (Large sphere diameter)³
Simplifying this equation using the given information, we have:
(Small sphere volume) / (128 cm³) = (18 cm)³ / (24 cm)³
Solving for the volume of the small sphere, we find that it is equal to 54 cm³.
