Calculate map length given scale and actual distance: a) 1:100, 7 km b) 1 cm = 0.5 km, 8 km

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1. Calculate map length

a) Scale 1:100, Actual distance = 7 km

Convert the actual distance from kilometres to centimetres.
$1 \text{ km} = 1000 \text{ m}$
$1 \text{ m} = 100 \text{ cm}$
So, $1 \text{ km} = 1000 \times 100 \text{ cm} = 100000 \text{ cm}$ .
Actual distance = $7 \text{ km} = 7 \times 100000 \text{ cm} = 700000 \text{ cm}$ .
Use the scale to find the map length.
A scale of $1:100$ means $1 \text{ cm}$ on the map represents $100 \text{ cm}$ in actual distance.
$\text{Map Length} = \frac{\text{Actual Distance}}{\text{Scale Factor}}$ $\text{Map Length} = \frac{700000 \text{ cm}}{100}$ $\text{Map Length} = 7000 \text{ cm}$
Convert the map length to meters if desired, but the question implies cm.
- The map length is $\boxed{\text{7000 cm}}$
b) Scale 1 cm = 0.5 km, Actual distance = 8 km
Use the given verbal scale.
The scale states that $1 \text{ cm}$ on the map represents $0.5 \text{ km}$ in actual distance.
Determine how many $0.5 \text{ km}$ segments are in $8 \text{ km}$ .
$\text{Number of segments} = \frac{8 \text{ km}}{0.5 \text{ km/cm}}$ $\text{Number of segments} = 16$
Multiply the number of segments by $1 \text{ cm}$ to find the map length.
$\text{Map Length} = 16 \times 1 \text{ cm}$ $\text{Map Length} = 16 \text{ cm}$
- The map length is $\boxed{\text{16 cm}}$
3. Calculate the suitable scale

a) Map length = 8 cm, Actual distance = 20 km
Convert the actual distance from kilometres to centimetres.
Actual distance = $20 \text{ km} = 20 \times 100000 \text{ cm} = 2000000 \text{ cm}$ .
Set up the ratio of map length to actual distance.
$\text{Scale} = \text{Map Length} : \text{Actual Distance}$ $\text{Scale} = 8 \text{ cm} : 2000000 \text{ cm}$
Simplify the ratio by dividing both sides by the map length ( $8 \text{ cm}$ ) to get a $1:X$ scale.
$\text{Scale} = \frac{8}{8} : \frac{2000000}{8}$ $\text{Scale} = 1 : 250000$
- The suitable scale is $\boxed{\text{1:250000}}$
b) Map length = 5.6 cm, Actual distance = 290 km
Convert the actual distance from kilometres to centimetres.
Actual distance = $290 \text{ km} = 290 \times 100000 \text{ cm} = 29000000 \text{ cm}$ .
Set up the ratio of map length to actual distance.
$\text{Scale} = \text{Map Length} : \text{Actual Distance}$ $\text{Scale} = 5.6 \text{ cm} : 29000000 \text{ cm}$
Simplify the ratio by dividing both sides by the map length ( $5.6 \text{ cm}$ ).
$\text{Scale} = \frac{5.6}{5.6} : \frac{29000000}{5.6}$ $\text{Scale} = 1 : 5178571.428...$
Round the scale factor to a reasonable whole number, typically for map scales.
$\text{Scale} \approx 1 : 5178571$
- The suitable scale is $\boxed{\text{1:5178571}}$
c) Map length = 4 cm, Actual distance = 2 km
Convert the actual distance from kilometres to centimetres.
Actual distance = $2 \text{ km} = 2 \times 100000 \text{ cm} = 200000 \text{ cm}$ .
Set up the ratio of map length to actual distance.
$\text{Scale} = \text{Map Length} : \text{Actual Distance}$ $\text{Scale} = 4 \text{ cm} : 200000 \text{ cm}$
Simplify the ratio by dividing both sides by the map length ( $4 \text{ cm}$ ).
$\text{Scale} = \frac{4}{4} : \frac{200000}{4}$ $\text{Scale} = 1 : 50000$
- The suitable scale is $\boxed{\text{1:50000}}$

Calculate map length given scale and actual distance: a) 1:100, 7 km b) 1 cm = 0.5 km, 8 km

Step-by-step solution

Convert the actual distance from kilometres to centimetres.

Use the scale to find the map length.

Convert the map length to meters if desired, but the question implies cm.

Use the given verbal scale.

Determine how many $0.5 \text{ km}$ segments are in $8 \text{ km}$ .

Multiply the number of segments by $1 \text{ cm}$ to find the map length.

Convert the actual distance from kilometres to centimetres.

Set up the ratio of map length to actual distance.

Simplify the ratio by dividing both sides by the map length ( $8 \text{ cm}$ ) to get a $1:X$ scale.

Convert the actual distance from kilometres to centimetres.

Set up the ratio of map length to actual distance.

Simplify the ratio by dividing both sides by the map length ( $5.6 \text{ cm}$ ).

Round the scale factor to a reasonable whole number, typically for map scales.

Convert the actual distance from kilometres to centimetres.

Set up the ratio of map length to actual distance.

Simplify the ratio by dividing both sides by the map length ( $4 \text{ cm}$ ).

Still stuck on this one?

More Mathematics Questions

Calculate map length given scale and actual distance: a) 1:100, 7 km b) 1 cm = 0.5 km, 8 km

Step-by-step solution

Convert the actual distance from kilometres to centimetres.

Use the scale to find the map length.

Convert the map length to meters if desired, but the question implies cm.

Use the given verbal scale.

Determine how many 0.5 km0.5 \text{ km}0.5 km segments are in 8 km8 \text{ km}8 km.

Multiply the number of segments by 1 cm1 \text{ cm}1 cm to find the map length.

Convert the actual distance from kilometres to centimetres.

Set up the ratio of map length to actual distance.

Simplify the ratio by dividing both sides by the map length (8 cm8 \text{ cm}8 cm) to get a 1:X1:X1:X scale.

Convert the actual distance from kilometres to centimetres.

Set up the ratio of map length to actual distance.

Simplify the ratio by dividing both sides by the map length (5.6 cm5.6 \text{ cm}5.6 cm).

Round the scale factor to a reasonable whole number, typically for map scales.

Convert the actual distance from kilometres to centimetres.

Set up the ratio of map length to actual distance.

Simplify the ratio by dividing both sides by the map length (4 cm4 \text{ cm}4 cm).

Still stuck on this one?

More Mathematics Questions

Determine how many $0.5 \text{ km}$ segments are in $8 \text{ km}$ .

Multiply the number of segments by $1 \text{ cm}$ to find the map length.

Simplify the ratio by dividing both sides by the map length ( $8 \text{ cm}$ ) to get a $1:X$ scale.

Simplify the ratio by dividing both sides by the map length ( $5.6 \text{ cm}$ ).

Simplify the ratio by dividing both sides by the map length ( $4 \text{ cm}$ ).