Problem restatement:
- We have a square plot with an area of $\(144 \mathrm{ft}^2\)$.
- A circular pool is to be installed inside this square plot.
- The pool is surrounded by a circular walkway that is 2 feet wide.
- The pool is to be as large as possible.
- We need to find the ratio:

$$
\(\frac{\text { Area of pool }}{\text { Area of walkway }}\)
$$


Step 1: Understand the dimensions of the plot and the pool
- Since the plot is square with area $144 \mathrm{ft}^2$ :

$$
\(\text { Side length of square }=\sqrt{144}=12 \text { feet }\)
$$

- The largest circular pool that fits inside a square will have a diameter equal to the side length of the square (since the circle fits snugly inside the square).
- So,

$$
\(\text { Diameter of the pool }+ \text { walkway }=12 \text { feet }\)
$$

- The outer radius (pool + walkway) is:

$$
\(R=\frac{12}{2}=6 \text { feet }\)
$$

- Since the walkway is 2 feet wide, the radius of the pool itself is:

$$
\(r=R-2=6-2=4 \text { feet }\)
$$

Step 2: Calculate the areas
- Area of the pool (circle with radius $r=4 \mathrm{ft}$ ):

$$
\(A_{\mathrm{pool}}=\pi r^2=\pi \times 4^2=16 \pi \mathrm{ft}^2\)
$$

- Area of the larger circle (pool + walkway) with radius $R=6 \mathrm{ft}$ :

$$
\(A_{\text {total }}=\pi R^2=\pi \times 6^2=36 \pi \mathrm{ft}^2\)
$$

- Area of the walkway is the total area minus the pool area:

$$
\(A_{\text {walkway }}=A_{\text {total }}-A_{\text {pool }}=36 \pi-16 \pi=20 \pi \mathrm{ft}^2\)
$$


Step 3: Calculate the ratio of the pool area to the walkway area

$$
\(\text { Ratio }=\frac{A_{\text {pool }}}{A_{\text {walkway }}}=\frac{16 \pi}{20 \pi}=\frac{16}{20}=\frac{4}{5}\)
$$


Final answer:

$$
\(\frac{4}{5}\)
$$


The ratio of the area of the pool to the area of the walkway is $\(\mathbf{4 : 5}\)$.