--- title: "Diagonal scale: BP highway 158 km" subject: "Surveying I" chapter: 1 key: "semester_3/surveying_i:diagonal-scale-bp-highway" source: https://www.bctengineer.com/q/semester_3/surveying_i/diagonal-scale-bp-highway site: BCT Engineering --- # Diagonal scale: BP highway 158 km Surveying I · Chapter 1: Introduction - Asked in: 2080 Bhadra, 5 marks - Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14 - Page: https://www.bctengineer.com/q/semester_3/surveying_i/diagonal-scale-bp-highway - Whole chapter: https://www.bctengineer.com/md/semester_3/surveying_i/chapter-1 ## Question The total length of BP highway is 158 km. In a highway map, 150 km out of 158 km is represented by a line 5 cm long. Find its R.F. Draw a diagonal scale to read single km and maximum 600 km. Show the distances 336 km and 457 km. ## Short answer RF is 1/3,000,000 and scale length is 20 cm. Divide into six 100 km parts of 3.333 cm, subdivide the left part into 10 km, and draw diagonals to read 1 km. Mark 336 km as 300+30+6 and 457 km as 400+50+7. ## Answer #### 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. ```text 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.** ## Check yourself The answer's key points as multiple-choice questions, with the correct option and why. 1. 150 km on ground is 5 cm on the map. The RF is A. 1/300,000 B. 1/3,000,000 C. 1/158,000 D. 1/30,000,000 Answer: B. 5/(1.5 x 10^7) = 1/3,000,000. (Key point: RF) 2. Scale length for 600 km is A. 20 cm B. 15 cm C. 6 cm D. 2 cm Answer: A. 600 x 10^5 / 3 x 10^6 = 20 cm. (Key point: Length) 3. One 100 km main division measures about A. 2.5 cm B. 5 cm C. 3.333 cm D. 0.333 cm Answer: C. 20 cm / 6 = 3.333 cm. (Key point: Main division) 4. 457 km is read using A. vertical at 400, diagonal from 5, 57th line B. vertical at 450, diagonal from 7, 5th line C. vertical at 500, diagonal from 43, 7th line D. vertical at 400, diagonal from 50, 7th horizontal line Answer: D. 457 = 400 + 50 + 7. 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