Step 1: Work with the radius first.
The Earth's radius is given as $6371$ km, so its full diameter, tip to tip, is $2\times6371=12742$ km. This is the real world size we must squeeze into the small model.
Step 2: Write the model as a fraction of the real Earth.
The model diameter of $45$ cm stands for the entire $12742$ km diameter. So every centimetre of the model carries a fixed number of kilometres, and we get that number by dividing the real size by the model size:
\[ 1 \text{ cm on model} = \frac{12742}{45} \text{ km} \]
Step 3: Carry out the division.
\[ \frac{12742}{45} = 283.1555 \text{ km} \]
Check: $45\times283=12735$, and the leftover $7$ km spread over $45$ gives $7/45=0.1556$, confirming the same figure.
Step 4: Round off as asked.
Keeping two digits after the decimal point,
\[ 283.1555 \approx 283.16 \text{ km} \]
Step 5: State the result.
So each centimetre on the model stands for close to $283.16$ km on the real Earth, which sits inside the expected band of $283.00$ to $283.25$ km.
$\boxed{283.16 \text{ km}}$