The sum of the volume of two solid spheres is $$\frac{1144}{3} cm^3$$. If the sum of their radii is 7 cm, then what will be the difference of the radii?

Let radii of two spheres be $$r_1$$ and $$r_2$$ cm respectively, => $$(r_1+r_2)=7$$ cm ---------------(i)

Sum of volume = $$\frac{4}{3}\pi (r_1)^3+\frac{4}{3}\pi (r_2)^3=\frac{1144}{3}$$

=> $$(\frac{4}{3}\times\frac{22}{7})[(r_1)^3+(r_2)^3]=\frac{1144}{3}$$

=> $$(r_1)^3+(r_2)^3=91$$ -------------(ii)

Cubing equation (i) on both sides, 

=> $$(r_1)^3+(r_2)^3+3(r_1)(r_2)(r_1+r_2)=343$$

Substituting value from equation (i) and (ii), we get :

=> $$21r_1r_2=343-91=252$$

=> $$r_1r_2=12$$ ---------------(iii)

Also, squaring equation (i) on both sides, => $$(r_1)^2+(r_2)^2+2(r_1)(r_2)=49$$

=> $$(r_1-r_2)^2+4r_1r_2=49$$

Substituting value from equation (iii),

=> $$(r_1-r_2)^2=49-48=1$$

=> $$(r_1-r_2)=1$$

=> Ans - (D)