Construct a diagonal scale of 30mm = 1km to read up to 6km in divisions of 100m.

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5. Construct a diagonal scale of $30\text{mm} = 1\text{km}$ to read up to $6\text{km}$ in divisions of $10\text{m}$.

• Step 1: Calculate the Representative Fraction (RF).

  $$\text{RF} = \frac{\text{Drawing length}}{\text{Actual length}} = \frac{30\text{mm}}{1\text{km}} = \frac{30\text{mm}}{1 \times 1000 \times 1000\text{mm}} = \frac{30}{1,000,000} = \frac{3}{100,000}$$

• Step 2: Calculate the total length of the scale. The scale needs to read up to $6\text{km}$.

  $$\text{Total length} = \text{RF} \times \text{Maximum length} = \frac{3}{100,000} \times 6\text{km} = \frac{3}{100,000} \times (6 \times 1000 \times 1000\text{mm}) = 180\text{mm}$$

• Step 3: Construct the main scale.

  1. Draw a line $180\text{mm}$ long.
  2. Divide this line into 6 equal primary divisions, each representing $1\text{km}$ (each division will be $30\text{mm}$ long).
  3. Label the divisions $0, 1, 2, 3, 4, 5\text{km}$ from left to right, starting with 0 at the first mark. The leftmost division (from 0 to the left) will be used for subdivisions.

• Step 4: Subdivide the leftmost primary division (for $100\text{m}$ divisions).

  1. The leftmost division represents $1\text{km}$ ($1000\text{m}$).
  2. Divide this $30\text{mm}$ segment into 10 equal parts. Each part will represent $100\text{m}$.
  3. Label these subdivisions $0, 100, 200, \dots, 900\text{m}$ from right to left.

• Step 5: Construct the diagonal part (for $10\text{m}$ divisions).

  1. Draw a vertical line upwards from the 0 mark on the main scale.
  2. Draw 10 equally spaced horizontal lines above the main scale, parallel to it.
  3. From the 0 mark on the main scale, draw a diagonal line to the 10th mark on the vertical line at the $1\text{km}$ mark (the left end of the first primary division).
  4. From each $100\text{m}$ subdivision mark (left of 0), draw vertical lines upwards to the top horizontal line.
  5. The intersection of these vertical lines with the diagonal lines will allow reading of $10\text{m}$ divisions.

• Indicate the following readings:

  • i) $5.81\text{km}$: Locate $5\text{km}$ on the main scale. Move left to the $800\text{m}$ mark on the main subdivision. Then, move up along the vertical line from $800\text{m}$ to the 1st diagonal line (representing $10\text{m}$).
  • ii) $2.53\text{km}$: Locate $2\text{km}$ on the main scale. Move left to the $500\text{m}$ mark on the main subdivision. Then, move up along the vertical line from $500\text{m}$ to the 3rd diagonal line (representing $30\text{m}$).
  • iii) $4.67\text{km}$: Locate $4\text{km}$ on the main scale. Move left to the $600\text{m}$ mark on the main subdivision. Then, move up along the vertical line from $600\text{m}$ to the 7th diagonal line (representing $70\text{m}$).

6. Construct a plain scale of $30\text{mm}$ represents $1\text{m}$ to read up to $5\text{m}$ in intervals of $100\text{mm}$.

• Step 1: Calculate the Representative Fraction (RF).

  $$\text{RF} = \frac{\text{Drawing length}}{\text{Actual length}} = \frac{30\text{mm}}{1\text{m}} = \frac{30\text{mm}}{1000\text{mm}} = \frac{3}{100}$$

• Step 2: Calculate the total length of the scale. The scale needs to read up to $5\text{m}$.

  $$\text{Total length} = \text{RF} \times \text{Maximum length} = \frac{3}{100} \times 5\text{m} = \frac{3}{100} \times (5 \times 1000\text{mm}) = 150\text{mm}$$

• Step 3: Construct the main scale.

  1. Draw a line $150\text{mm}$ long.
  2. Divide this line into 5 equal primary divisions, each representing $1\text{m}$ (each division will be $30\text{mm}$ long).
  3. Label the divisions $0, 1, 2, 3, 4\text{m}$ from left to right, starting with 0 at the first mark. The leftmost division (from 0 to the left) will be used for subdivisions.

• Step 4: Subdivide the leftmost primary division (for $100\text{mm}$ divisions).

  1. The leftmost division represents $1\text{m}$ ($1000\text{mm}$).
  2. Divide this $30\text{mm}$ segment into 10 equal parts. Each part will represent $100\text{mm}$ (or $0.1\text{m}$).
  3. Label these subdivisions $0, 100, 200, \dots, 900\text{mm}$ (or $0.1, 0.2, \dots, 0.9\text{m}$) from right to left.

• Indicate the following readings:

  • i) $2.5\text{m}$: Locate $2\text{m}$ on the main scale. Move left to the $500\text{mm}$ (or $0.5\text{m}$) mark on the subdivision.
  • ii) $4.5\text{m}$: Locate $4\text{m}$ on the main scale. Move left to