This is a geometry problem involving density and the volume of a sphere in a creative way. The fact that the two spheres are the same metal mean that we can figure out the density of the metal and from that, find the volume of the 7 kg sphere. When the two spheres are melted together, the new sphere will have the combined volume of the two spheres, allowing us to find the radius of the new big sphere.
First, find the volume of sphere 1:
Sphere 1:
mass = 1 kg
radius = 3 cm
volume = 4/3 pi r^3 = 4/3 pi (3 cm )^3 = 36*pi cm^3= 113.1 cm^3
Then, find the density of the material from sphere 1:
density = mass/volume = 1 kg/36pi cm^3 = 1/(36*pi) kg/cm^3
This is the density of the metal. From this density, we can find the volume of sphere 2
Sphere 2
mass = 7 kg
volume = mass/density = 7 kg / [1/(36*pi) kg/cm^3]=791.7 cm^3
Thus, the volume of the single big sphere is the volume of the two melted spheres combined:
Total vol = vol sphere 1+ vol sphere 2 =113.1 + 791.7 = 904.8 cm^3
From this volume, the new diameter of the sphere can be found by solving for the radius via the volume equation (and taking the cubed root):
radius = (3* tot Vol/4)^(1/3) = (3*904.8/4)^(1/3)=8.78 cm
The diameter is then twice the radius:
diameter=2r=17.6 cm
