An area of 324 cm^2 on a map represents an area of 20.25 km^2.

1. Set up the proportional relationship between actual area and map area.

$20.25\text{ km}^2$ is represented by $324\text{ cm}^2$ on the map.

For $81\text{ km}^2$, map area
$=324\times \frac{81}{20.25}$

2. Calculate the map area for the resort.

$324\times \frac{81}{20.25}=324\times 4=1296$

So the area on the map is $1296\text{ cm}^2$.

1. Find the linear scale corresponding to the given area scale.

$324\text{ cm}^2$ on the map represents $20.25\text{ km}^2$ in actual area.

Taking square roots,
$\sqrt{324}\text{ cm}=\sqrt{20.25}\text{ km}$
$18\text{ cm}=4.5\text{ km}$

2. Use the linear scale to find the actual distance represented by 54 cm.

Since $18\text{ cm}$ represents $4.5\text{ km}$,
$54\text{ cm}=3\times18\text{ cm}$
Actual distance
$=3\times4.5=13.5\text{ km}$

1. Find the linear scale and express it as $1:n$ with the same units.

$324\text{ cm}^2$ on the map represents $20.25\text{ km}^2$ in actual area.

Taking square roots,
$18\text{ cm} \text{ on the map represents } 4.5\text{ km actual}.$
So
$1\text{ cm} \text{ on the map represents } \frac{4.5}{18}=0.25\text{ km actual}.$
Convert to centimetres:
$0.25\text{ km}=0.25\times 100000=25000\text{ cm}.$
Hence the scale is
$1:25000.$