The relationship between the surface area, diameter, and volume of spheres can be derived using the standard formulas:

• Surface area of a sphere: $S = 4\pi r^2$
• Volume of a sphere: $V = \frac{4}{3}\pi r^3$
• Diameter: $d = 2r$

(a) Ratio of diameters

Let the radii of spheres A and B be $r_A$ and $r_B$, respectively.

If the surface area of sphere A is 16 times that of sphere B: $S_A = 16S_B \quad \text{or} \quad 4\pi r_A^2 = 16(4\pi r_B^2)$

Simplifying: $r_A^2 = 16r_B^2 \quad \Rightarrow \quad r_A = 4r_B$

Since diameter $d = 2r$: $\frac{d_A}{d_B} = \frac{2r_A}{2r_B} = \frac{r_A}{r_B} = 4$

Thus, the ratio of the diameters is: $\boxed{4:1}$

---

(b) Ratio of volumes

Using the volume formula $V = \frac{4}{3}\pi r^3$, the ratio of the volumes is: $\frac{V_A}{V_B} = \frac{\frac{4}{3}\pi r_A^3}{\frac{4}{3}\pi r_B^3} = \frac{r_A^3}{r_B^3}$

From part (a), $r_A = 4r_B$: $\frac{V_A}{V_B} = \frac{(4r_B)^3}{r_B^3} = \frac{64r_B^3}{r_B^3} = 64$

Thus, the ratio of the volumes is: $\boxed{64:1}$

---

Summary of Results:

(a) The ratio of diameters is $\boxed{4:1}$.
(b) The ratio of volumes is $\boxed{64:1}$.

Would you like further explanations or assistance? Here are 5 related questions to deepen your understanding:

1. What happens to the surface area if the diameter of a sphere is doubled?
2. How does the ratio of volumes change if the ratio of radii is $3:1$?
3. Derive the formula for surface area in terms of diameter rather than radius.
4. How would you calculate the ratio of circumferences for the two spheres?
5. If sphere B has a radius of 5 cm, what is the radius of sphere A?

Tip: Remember that surface area scales with the square of the radius, while volume scales with the cube of the radius!