To solve this problem, let's break it down into steps.

1. Identify the dimensions of the building and the walkway:

   • The building has a rectangular footprint of $100 \text{ ft} \times 150 \text{ ft}$.
   • There is a walkway of width $x$ feet surrounding the building on all four sides.

2. Calculate the area of the building alone:

   • The area of the building is: $100 \times 150 = 15,000 \text{ square feet}$

3. Set up the equation for the total area including the walkway:

   • With the walkway, the dimensions of the entire area become:
     • Width: $100 + 2x$
     • Length: $150 + 2x$
   • The total area, including the walkway, is given to be 16,536 square feet. Therefore, we can set up the equation: $(100 + 2x)(150 + 2x) = 16,536$

4. Expand and solve the equation:

   • Expanding the left side: $100 \times 150 + 100 \times 2x + 150 \times 2x + 2x \times 2x = 16,536$ $15,000 + 200x + 300x + 4x^2 = 16,536$ $4x^2 + 500x + 15,000 = 16,536$
   • Simplify the equation by moving 16,536 to the left side: $4x^2 + 500x + 15,000 - 16,536 = 0$ $4x^2 + 500x - 1,536 = 0$
   • Divide the equation by 4 for simplicity: $x^2 + 125x - 384 = 0$

5. Solve the quadratic equation $x^2 + 125x - 384 = 0$:

   • Using the quadratic formula: $x = \frac{-125 \pm \sqrt{125^2 + 4 \times 384}}{2 \times 1}$ $x = \frac{-125 \pm \sqrt{15625 + 1536}}{2}$ $x = \frac{-125 \pm \sqrt{17161}}{2}$ $x = \frac{-125 \pm 131}{2}$
   • Solving this, we get two values for $x$: $x = \frac{6}{2} = 3 \quad \text{or} \quad x = \frac{-256}{2} = -128$
   • Since $x$ represents a width, it must be positive, so $x = 3$.

Conclusion:

The width of the walkway is $\mathbf{3 \text{ feet}}$.

Would you like more details or have any questions? Here are some related questions:

1. How would the width of the walkway change if the building had a different footprint?
2. What if the total area were different, say 18,000 square feet? How would we solve for $x$?
3. Can you derive a general formula for the width of the walkway based on the building dimensions and total area?
4. What happens to the dimensions of the walkway if we have an irregular-shaped building?
5. How would this problem change if there were a circular building instead of a rectangular one?

Tip: Always check if your answer is reasonable by plugging it back into the original problem to verify it meets all given conditions.