Diagonal scale: BP highway 158 km

RF

150 km on the ground $= 150 \times 10^5$ cm $= 1.5 \times 10^7$ cm is shown by 5 cm.

$$\text{RF} = \frac{5}{1.5 \times 10^{7}} = \frac{1}{3{,}000{,}000}$$

Length of the scale

$$\text{Length for 600 km} = \frac{600 \times 10^5}{3 \times 10^6} = 20\ \text{cm}$$

100 km $= 3.333$ cm; 10 km $= 0.3333$ cm; 1 km $= 0.0333$ cm.

Construction

1. Draw a line 20 cm long and divide it into 6 equal main divisions of 3.333 cm (100 km each). Mark 0 at the end of the first division and 100, 200, ... 500 on the others.
2. Divide the first division into 10 equal parts of 0.333 cm (10 km each); number 10 to 90 to the left of 0.
3. Draw a rectangle of height about 3 cm on the scale, divide the height into 10 equal parts, and draw horizontal lines.
4. Draw the diagonals: join the top of the sub-division marked 10 to the bottom 0, and the others parallel to it. Each horizontal line cuts a diagonal at $\tfrac{1}{10}$ of 10 km $=$ 1 km steps.

 10 |  /  /  /  /  /  /  /  /  /  /  |
  5 | /  /                            |
  0 +----+----+----+----+----+----+---+
   90..0  100  200  300  400  500 (km)

Distances

• 336 km $= 300 + 30 + 6$: vertical at 300, diagonal from sub-division 30, 6th horizontal line. The intercept from this point to the 300 vertical is 336 km.
• 457 km $= 400 + 50 + 7$: vertical at 400, diagonal from sub-division 50, 7th horizontal line.

Answer: RF = 1/3,000,000; scale length = 20 cm.