To find the greatest possible area of each swimming pool, we'll use the formulas for the area of a rectangle, which is:
\[ \text{Area} = \text{Length} \times \text{Width} \]
For John's pool:
- Length = 3 yards
- Width can be no more than 12 yards.
Thus, the area for John's pool can be calculated as:
\[ \text{Area}{John} = 3 , \text{yards} \times \text{Width}{John} \]
To maximize John's area, we take the maximum width:
\[ \text{Width}_{John} = 12 , \text{yards} \]
So:
\[ \text{Area}_{John} = 3 , \text{yards} \times 12 , \text{yards} = 36 , \text{square yards} \]
For Andy's pool:
- Length = 4 yards
- Width can be no more than 8 yards.
Thus, the area for Andy's pool can be calculated as:
\[ \text{Area}{Andy} = 4 , \text{yards} \times \text{Width}{Andy} \]
To maximize Andy's area, we take the maximum width:
\[ \text{Width}_{Andy} = 8 , \text{yards} \]
So:
\[ \text{Area}_{Andy} = 4 , \text{yards} \times 8 , \text{yards} = 32 , \text{square yards} \]
However, we want both pools to have the same area. Currently, John's pool has an area of 36 square yards, and Andy's pool has an area of 32 square yards. They cannot build their pools with the maximum widths if they are to have the same area.
To find the shared area, we can calculate a common area that fits both constraints.
Let’s denote the common area as \( A \).
For John’s pool, using the length of 3 yards:
\[ A = 3 , \text{yards} \times \text{Width}{John} \quad \Rightarrow \quad \text{Width}{John} = \frac{A}{3} \]
For Andy's pool, using the length of 4 yards:
\[ A = 4 , \text{yards} \times \text{Width}{Andy} \quad \Rightarrow \quad \text{Width}{Andy} = \frac{A}{4} \]
Now using the width limits, we have:
For John’s pool:
\[ \frac{A}{3} \leq 12 \quad \Rightarrow \quad A \leq 36 \]
For Andy’s pool:
\[ \frac{A}{4} \leq 8 \quad \Rightarrow \quad A \leq 32 \]
The limiting area is \( A = 32 \) square yards, which is the greatest possible area that both pools can have while adhering to all constraints.
Thus, the greatest possible area of each of their pools can be:
\[ \text{Greatest area for both pools} = \boxed{32} , \text{square yards.} \]