To solve this problem, we will use the scale provided and apply it to both length and area.

a. Calculate the actual length of the island

The scale is given as $1:125,000$, meaning 1 cm on the map corresponds to 125,000 cm in reality.

• Length on the map = 9.4 cm
• Actual length = $9.4 \times 125,000$ cm
• Convert cm to kilometers (since $100,000$ cm = 1 km)

$\text{Actual length} = \frac{9.4 \times 125,000}{100,000} \, \text{km} = 11.75 \, \text{km}$

So, the actual length of the island is 11.75 km.

b. Calculate the actual area of the region

For area, the scale factor needs to be squared because both dimensions are scaled.

• Area on the map = 6.4 cm²
• The scale factor is $125,000$, so the area scale factor is $(125,000)^2$

$\text{Actual area} = 6.4 \times (125,000)^2 \, \text{cm}^2$

• Convert from cm² to km² (since $1 \, \text{km}^2 = (100,000)^2 \, \text{cm}^2$)

$\text{Actual area} = \frac{6.4 \times (125,000)^2}{(100,000)^2} \, \text{km}^2$

Let me calculate that for you.

The actual area of the region is 10 km².

Summary:

• a. The actual length of the island is 11.75 km.
• b. The actual area of the region is 10 km².

Do you need more details or have any further questions?

Here are 5 related questions to expand this concept:

1. If the length on the map were doubled, what would the actual length of the island be?
2. How would you calculate the perimeter of an island given its area and scale?
3. What is the relationship between map scale and distance conversion?
4. How can we calculate volume if height data is also provided on the map?
5. How would the calculation change if the map scale were 1:50,000 instead of 1:125,000?

Tip: When converting areas on a map, always remember to square the scale factor before applying it to the area.