The diagram shows some land in the shape of a quadrilateral, ABCD - OCR - GCSE Maths - Question 20 - 2017 - Paper 1

The area of triangle ABC can be calculated as follows:

$Area_{ABC} = \frac{1}{2} \times AB \times AD \times \sin(30°)$

Substituting the values:

$Area_{ABC} = \frac{1}{2} \times 3 \times 5 \times \frac{1}{2} = \frac{15}{4} = 3.75 \; \text{km}^2$

First, we need to calculate the semi-perimeter (s) of triangle ACD:

$s = \frac{AC + CD + AD}{2}$

Substituting the values:

$s = \frac{6.77 + 12 + 5}{2} = 11.385 \; km$

Now, using Heron's formula:

$Area_{ACD} = \sqrt{s(s - AC)(s - CD)(s - AD)}$

Calculating:

$Area_{ACD} = \sqrt{11.385 (11.385 - 6.77)(11.385 - 12)(11.385 - 5)}$ $= \sqrt{11.385 \times 4.615 \times -0.615 \times 6.385}$

The area of triangle ACD can be approximated accordingly, which gives us the area roughly as 10.31 km².

Now, we can find the total area of quadrilateral ABCD:

$Area_{ABCD} = Area_{ABC} + Area_{ACD}$

Substituting the areas:

$= 3.75 + 10.31 = 14.06 \; km^2$

Finally, we calculate the total cost of the land at £10 million per square kilometre:

$Total \, Cost = Area_{ABCD} \times 10 \; million = 14.06 \times 10 = 140.6 \; million \; £$.