Given: A closed rectangular box  

Step 1: Surface area of the box  
2(lb + lh + bh) = 846  

Step 2: Sum of the lengths of all edges  
4(l + b + h) = 144  
=> l + b + h = 144 / 4 = 36  …(1)  

Step 3: Square equation (1)  
(l + b + h)^2 = l^2 + b^2 + h^2 + 2(lb + lh + bh) = 36^2 = 1296  
l^2 + b^2 + h^2 = 1296 − 846 = 450  

Step 4: Diameter of the sphere (space diagonal of the box)  
d = √(l^2 + b^2 + h^2) = √450 = 15√2 = 2r  

Step 5: Radius of the sphere  
r = (15√2) / 2 = 15/√2  

Step 6: Volume of the sphere  
V = (4/3) π r^3 = (4/3) π (15√2 / 2)^3  
V = (4/3) π * 1125√2 / 8  
V = 1125 π √2 / 6  
V = 1125 π √2