1. **State the problem:** We have two similar right circular cylinders A and B. Cylinder A has volume $112\pi$ and radius 4 units. Cylinder B has volume $3024\pi$. We know the surface areas are $k\pi$ for A and $n\pi$ for B. We need to find $n - k$. 2. **Recall formulas:** Volume of a cylinder: $$V = \pi r^2 h$$ Surface area of a cylinder: $$S = 2\pi r^2 + 2\pi r h$$ 3. **Find height of cylinder A:** Given $V_A = 112\pi$, $r_A = 4$. $$112\pi = \pi \times 4^2 \times h_A$$ $$112\pi = \pi \times 16 \times h_A$$ Divide both sides by $\pi$: $$112 = 16 h_A$$ $$\cancel{16} h_A = \cancel{112}$$ $$h_A = \frac{112}{16} = 7$$ 4. **Calculate surface area of cylinder A:** $$S_A = 2\pi (4)^2 + 2\pi (4)(7) = 2\pi (16) + 2\pi (28) = 32\pi + 56\pi = 88\pi$$ So, $k = 88$. 5. **Find scale factor between cylinders:** Since cylinders are similar, linear scale factor $s$ satisfies: $$\frac{V_B}{V_A} = s^3$$ $$\frac{3024\pi}{112\pi} = s^3$$ $$\frac{3024}{112} = s^3$$ Simplify fraction: $$\frac{3024}{112} = 27$$ So, $$s^3 = 27 \implies s = 3$$ 6. **Find radius and height of cylinder B:** $$r_B = s \times r_A = 3 \times 4 = 12$$ $$h_B = s \times h_A = 3 \times 7 = 21$$ 7. **Calculate surface area of cylinder B:** $$S_B = 2\pi (12)^2 + 2\pi (12)(21) = 2\pi (144) + 2\pi (252) = 288\pi + 504\pi = 792\pi$$ So, $n = 792$. 8. **Find $n - k$:** $$n - k = 792 - 88 = 704$$ **Final answer:** $$\boxed{704}$$