Question

Stan and Olga are considering buying a sheep farm. A surveyorhas supplied them with the given accurate sketch. Find the areaof the property, giving your answer in:km2

Answer 1

The approximate area of the property is;

`75km^2`

Here, we want to calculate the area of the property

Firstly, we can see that the farm is divided into two shapes (two triangles actually)

By finding the areas of the two triangles, adding tehm up, we will have the area of the farm

Firstly, we need to get all the sides of the triangles

Let us start with the triangle below

It is simply the length of the diagonal QS

We can find this by the use of cosine rule

We have this as;

`\begin{gathered} p^2=q^2+s^2-2qsCosP \\ p^2=8^2+12^2\text{ -2(8)(12)Cos70} \\ p^2\text{ = 64 + 144 - 65.67} \\ p\text{ = 11.93 km} \end{gathered}`

Now, by using this calculated diagonal, we can get the missing side of the top triangle

We have the missing side calculated as follows;

`\begin{gathered} q^2=r^2+s^2-2rsCosQ \\ q^2=11.93^2+10^2-2(11.93)(10)Cos30 \\ q^2\text{ = 35.69} \\ \text{q = 5.97 km} \end{gathered}`

So, we now use Heron's formula for each of the two triangles before adding up

For triangle QPS, we have;

`\begin{gathered} A\text{ = }\sqrt[]{s(s-a)(s-b)(s-c)} \\ a,b\text{ and c are the sides} \\ \text{s = }(a+b+c)/(2)\text{ = }(8+12+11.93)/(2)\text{ = 15.965} \\ \\ A\text{ = }\sqrt[]{15.965(15.965-8)(15.965-12)(15.965-11.93)} \\ A\text{ = }\sqrt[]{15.965(7.965)(3.965)(4.035)} \\ A=45.105km^2 \end{gathered}`

For triangle QRS, we have;

`\begin{gathered} A\text{ = }\sqrt[]{s(s-a)(s-b)(s-c)} \\ a,b,c\text{ are the sides} \\ s\text{ = }(a+b+c)/(2)\text{ = }\frac{10\text{ + 5.97+11.93}}{2}\text{ = 13.95} \\ \\ A\text{ = }\sqrt[]{13.95(13.95-10)(13.95-5.97)(13.95-11.93)} \\ A=29.803km^2 \end{gathered}`

The area of the sheep farm is the sum of both areas

We have this as;

`\text{Total area = 29.803 + 45.105 = 74.908 km}^2`