The problem describes a circular vegetable garden with a diameter of 24.0 meters, a flowerbed with a diameter of 6.0 meters, and circular paths 1.5 meters wide. We need to write the equations for all the circles that Don needs to draw, assuming the center of the garden is at $(0, 0)$.

Step 1: Equation for the Outer Circle of the Garden

The diameter of the circular garden is 24.0 meters, so the radius is: $r = \frac{24.0}{2} = 12.0 \text{ meters}$ The equation of a circle centered at $(0, 0)$ with radius $r$ is: $x^2 + y^2 = r^2$ Substituting $r = 12.0$: $x^2 + y^2 = 12^2 = 144$ Thus, the equation of the outer circle is: $x^2 + y^2 = 144$

Step 2: Equation for the Flowerbed Circle

The diameter of the flowerbed is 6.0 meters, so the radius is: $r = \frac{6.0}{2} = 3.0 \text{ meters}$ The equation of the circle for the flowerbed is: $x^2 + y^2 = r^2$ Substituting $r = 3.0$: $x^2 + y^2 = 3^2 = 9$ Thus, the equation of the flowerbed circle is: $x^2 + y^2 = 9$

Step 3: Equation for the Inner Path

The paths around the garden have a uniform width of 1.5 meters. The radius of the inner path is: $r = 12.0 - 1.5 = 10.5 \text{ meters}$ The equation of the circle for the inner path is: $x^2 + y^2 = r^2$ Substituting $r = 10.5$: $x^2 + y^2 = 10.5^2 = 110.25$ Thus, the equation of the inner path circle is: $x^2 + y^2 = 110.25$

Step 4: Equation for the Outer Path

The outer edge of the path extends 1.5 meters beyond the inner edge. The radius of the outer path is: $r = 12.0 + 1.5 = 13.5 \text{ meters}$ The equation of the circle for the outer path is: $x^2 + y^2 = r^2$ Substituting $r = 13.5$: $x^2 + y^2 = 13.5^2 = 182.25$ Thus, the equation of the outer path circle is: $x^2 + y^2 = 182.25$

Step 5: Equation for the Middle Circle

Between the flowerbed and the outer circle, Don could add another circle for symmetry. We can assume its radius is the average of the flowerbed radius (3.0 meters) and the outer circle radius (12.0 meters): $r = \frac{12.0 + 3.0}{2} = 7.5 \text{ meters}$ The equation for this middle circle is: $x^2 + y^2 = 7.5^2 = 56.25$ Thus, the equation of the middle circle is: $x^2 + y^2 = 56.25$

Summary of Equations:

1. Outer garden circle: $x^2 + y^2 = 144$
2. Flowerbed circle: $x^2 + y^2 = 9$
3. Inner path circle: $x^2 + y^2 = 110.25$
4. Outer path circle: $x^2 + y^2 = 182.25$
5. Middle circle (optional for symmetry): $x^2 + y^2 = 56.25$

Would you like further details or explanations on any of these steps?

Follow-up Questions:

1. How would the equations change if the garden were shifted from the origin $(0,0)$?
2. How does the width of the path influence the equations?
3. Could you add more concentric circles in the design, and what would their equations be?
4. What is the area of each of the circles mentioned above?
5. How could you modify this design to include elliptical shapes instead of circles?

Tip: When dealing with circular objects, always remember that the radius is half of the diameter, and it directly influences the equation of the circle.