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A road is 8.4 cm long on a map with a scale of 1:50,000. A lake has an area of 2.5 cm² on the same map. Find the actual length of the road and the actual area of the lake.

Mathematics

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3 parts
a

4.2 km\text{4.2 km}

b

0.625 km2\text{0.625 km}^2

c

verified\text{verified}

Step-by-step solution

8 steps

Here's the solution to Question 3:

Part a· 2 steps

To find the actual length of the road, we use the given map scale and convert the units.

  1. Calculate the actual length in centimeters.

    The scale is 1:50 0001 : 50\,000, meaning 1 cm1 \text{ cm} on the map represents 50 000 cm50\,000 \text{ cm} in reality.

    Actual length=8.4 cm×50 000\text{Actual length} = 8.4 \text{ cm} \times 50\,000 Actual length=420 000 cm\text{Actual length} = 420\,000 \text{ cm}

  2. Convert the actual length from centimeters to kilometers.

    Given 1 km=100 000 cm1 \text{ km} = 100\,000 \text{ cm}.

    Actual length in km=420 000 cm100 000 cm/km\text{Actual length in km} = \frac{420\,000 \text{ cm}}{100\,000 \text{ cm/km}} Actual length in km=4.2 km\text{Actual length in km} = 4.2 \text{ km}

    The actual length of the road is 4.2 km\boxed{\text{4.2 km}}.

Part b

0.625 km2\text{0.625 km}^2

To find the actual area of the lake, we use the area scale factor, which is the square of the length scale factor.

  1. Calculate the area scale factor.

    The length scale factor is 50 00050\,000.

    Area scale factor=(Length scale factor)2=(50 000)2\text{Area scale factor} = (\text{Length scale factor})^2 = (50\,000)^2 Area scale factor=2 500 000 000\text{Area scale factor} = 2\,500\,000\,000

  2. Calculate the actual area in square centimeters.

    Actual area=2.5 cm2×2 500 000 000\text{Actual area} = 2.5 \text{ cm}^2 \times 2\,500\,000\,000 Actual area=6 250 000 000 cm2\text{Actual area} = 6\,250\,000\,000 \text{ cm}^2

  3. Convert the actual area from square centimeters to square kilometers.

    Since 1 km=100 000 cm1 \text{ km} = 100\,000 \text{ cm}, then 1 km2=(100 000 cm)2=10 000 000 000 cm21 \text{ km}^2 = (100\,000 \text{ cm})^2 = 10\,000\,000\,000 \text{ cm}^2.

    Actual area in km2=6 250 000 000 cm210 000 000 000 cm2/km2\text{Actual area in km}^2 = \frac{6\,250\,000\,000 \text{ cm}^2}{10\,000\,000\,000 \text{ cm}^2/\text{km}^2} Actual area in km2=0.625 km2\text{Actual area in km}^2 = 0.625 \text{ km}^2

    The actual area of the lake is 0.625 km2\boxed{\text{0.625 km}^2}.

Part c

verified\text{verified}

To verify the given areas, we will convert the actual area to square centimeters and compare it with the map area using the area scale factor.

  1. Convert the actual area from square meters to square centimeters.

    Given actual area is 125 000 m2125\,000 \text{ m}^2.
    Since 1 m=100 cm1 \text{ m} = 100 \text{ cm}, then 1 m2=(100 cm)2=10 000 cm21 \text{ m}^2 = (100 \text{ cm})^2 = 10\,000 \text{ cm}^2.

    Actual area in cm2=125 000 m2×10 000 cm2/m2\text{Actual area in cm}^2 = 125\,000 \text{ m}^2 \times 10\,000 \text{ cm}^2/\text{m}^2 Actual area in cm2=1 250 000 000 cm2\text{Actual area in cm}^2 = 1\,250\,000\,000 \text{ cm}^2

  2. Calculate the actual area based on the map area and the area scale factor.

    The map area is 0.5 cm20.5 \text{ cm}^2. The area scale factor is (50 000)2=2 500 000 000(50\,000)^2 = 2\,500\,000\,000.

    Calculated actual area=0.5 cm2×2 500 000 000\text{Calculated actual area} = 0.5 \text{ cm}^2 \times 2\,500\,000\,000 Calculated actual area=1 250 000 000 cm2\text{Calculated actual area} = 1\,250\,000\,000 \text{ cm}^2

  3. Compare the calculated actual area with the given actual area.

    The calculated actual area (1 250 000 000 cm21\,250\,000\,000 \text{ cm}^2) matches the given actual area (1 250 000 000 cm21\,250\,000\,000 \text{ cm}^2).
    Therefore, the statement is verified\boxed{\text{verified}}.

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