Final answer:
To find the radius of the sphere formed by melting 54 solid hemispheres, we need to find the volume of the resulting sphere and then calculate its radius. The radius of each solid hemisphere is 1 cm. The radius of the sphere formed by melting the 54 hemispheres is 3 cm.
Step-by-step explanation:
To find the radius of the sphere formed by melting 54 solid hemispheres, we need to find the volume of the resulting sphere and then calculate its radius.
The formula for the volume of a sphere is V = (4/3)πr³, where r is the radius of the sphere.
Since each hemisphere has a diameter of 2 cm, the radius of each hemisphere is 1 cm. Therefore, the radius of each solid hemisphere is 1 cm.
The volume of each solid hemisphere is (2/3)π(1^3) = (2/3)π cm³.
The total volume of the 54 solid hemispheres is (54)(2/3)π cm³ = 36π cm³.
Therefore, the volume of the resulting sphere is 36π cm³.
Using the formula for the volume of a sphere, we can equate the volume to 36π and solve for the radius:
(4/3)πr³ = 36π
r³ = 36/((4/3)π)
r³ = 36/(4/3)
r³ = 36 x (3/4)
r³ = 27
r = ∛27
r = 3 cm.
Therefore, the radius of the sphere formed by melting 54 solid hemispheres is 3 cm.