To solve this problem, let's extract the key points:

We are given:

• The formula for $r$:
  $r = \sqrt{\frac{A + \frac{1}{8\pi}}{8\pi}} - 0.5$ where $r$ is the radius in meters, and $A$ represents the square meters of laminate required.
• $r = 3$ meters is given.
• We need to find the corresponding value of $A$, the total surface area.

Step-by-Step Solution:

1. Substitute $r = 3$ into the formula: $3 = \sqrt{\frac{A + \frac{1}{8\pi}}{8\pi}} - 0.5$

2. Rearrange to isolate the square root term: $3 + 0.5 = \sqrt{\frac{A + \frac{1}{8\pi}}{8\pi}}$ $3.5 = \sqrt{\frac{A + \frac{1}{8\pi}}{8\pi}}$

3. Square both sides to eliminate the square root: $(3.5)^2 = \frac{A + \frac{1}{8\pi}}{8\pi}$ $12.25 = \frac{A + \frac{1}{8\pi}}{8\pi}$

4. Multiply through by $8\pi$ to simplify: $12.25 \cdot 8\pi = A + \frac{1}{8\pi}$ $98\pi = A + \frac{1}{8\pi}$

5. Subtract $\frac{1}{8\pi}$: $A = 98\pi - \frac{1}{8\pi}$

6. Calculate the terms numerically:

   • $98\pi \approx 98 \cdot 3.1416 = 307.88$
   • $\frac{1}{8\pi} \approx \frac{1}{8 \cdot 3.1416} = \frac{1}{25.1328} \approx 0.0398$
   • $A = 307.88 - 0.0398 = 307.84 \, \text{square meters}$

Compare with Options:

The value $A = 307.84$ does not match any of the given options exactly. However, **the process and result indicate the correct method to evaluate this problem. Let me know if you'd like further assistance to resolve any inconsistencies in the options provided.