Figure 2 shows the design for a triangular garden ABC where AB = 7 m, AC = 13 m and BC = 10 m - Edexcel - A-Level Maths Pure - Question 1 - 2013 - Paper 5

First, we need to calculate the area of the shaded region S.

The area of triangle ABC can be found using the formula:

$$\text{Area} = \frac{1}{2} \times AB \times AC \times \sin(θ)$$

Substituting the previously found θ:

$$\text{Area}_{ABC} = \frac{1}{2} \times 7 \times 13 \times \sin(0.865)$$

Calculating: $ext{Area}_{ABC} \approx \frac{1}{2} \times 7 \times 13 \times 0.755\approx 34.6\,m^2$

Next, we need to find the area of sector ABD:

$$\text{Area}_{ABD} = \frac{1}{2} \times r^2 \times θ = \frac{1}{2} \times 7^2 \times 0.865$$

Calculating this gives: $ext{Area}_{ABD} \approx \frac{1}{2} \times 49 \times 0.865 \approx 21.2\,m^2$

Now the area of the shaded region S can be calculated as:

\text{Area}_{S} = \text{Area}_{ABC} - \text{Area}_{ABD}\ approx 34.6 - 21.2 = 13.4\,m^2$$ Given that 50 g of grass seed are needed per square metre:

\text{Total seed needed} = 13.4 \times 50 = 670,g$$

Rounding to the nearest 10 g, the answer is 670 g.