The diagram below shows a cyclic quadrilateral TSMR - NSC Technical Mathematics - Question 6 - 2024 - Paper 2

To find ∠M, we use the fact that the sum of angles in a triangle is 180°:

$$ ∠M = 180° - ∠T - ∠R₁ = 180° - 67° - 42.5° = 110.5° $.

Using the sine rule:

$\frac{SM}{\sin R_1} = \frac{SR}{\sin R}$

Where:

• $R_1 = 42.5°$
• $SR ≈ 22.33 m$
• $R ≈ 110.5°$

Using the sine rule:

$SM = \frac{SR \times \sin R_1}{\sin R}$

Replacing with values:

$SM = \frac{22.33 \times \sin(42.5°)}{\sin(110.5°)}$

Calculating this gives:

$SM ≈ 16.39 m$

The area of triangle ΔSMR can be calculated using the formula:

$Area = \frac{1}{2} \times SR \times SM \times \sin(∠M)$

Thus:

$Area = \frac{1}{2} \times 22.33 \times 16.39 \times \sin(110.5°)$

This yields an area of approximately 75.89 m².

Given that one bag of fertiliser covers 15,178 m², the number of bags needed:

$$ \text{Number of Bags} = \frac{Area}{\text{Coverage per Bag}} = \frac{75.89}{15178} \approx 0.005 $,

which means 0 bags of fertiliser are necessary.