Question

An ornamental lawn has the shape of a triangle.Two of the sides have lengths 18 m and 24 m, and the angle between these twosides is 72°.The perimeter of the triangle is to be marked with a low fence.a)Find the total length of fencing required, giving the answer to 3 significantfigures.b) Find the area of the ornamental lawn, giving the answer to 3 significant figures.

Answer 1

a) 67.2 m

b) 205 m²

Step-by-step explanation:

a) Given: We have two sides and an included angle of a triangle. To find the third side of the triangle, we will apply cosine rule:

`c^2=a^2+b^2\text{ -2abcosC}`

let a = 18m, b = 24m, c = ?

C = included angle = 72°

Substituting the values in the formula:

`\begin{gathered} c^2=18^2+24^2\text{ -2(18)(24)(cos72)} \\ c^2\text{ = 324 + 576 - 864(0.3090)} \\ c^2\text{ = 900 - 266.976 = 633.024} \\ c\text{ = }\sqrt[]{633.024}\text{ } \\ c\text{ = 25.16 m} \end{gathered}`

To find the perimeter of the triangle, we will add all three sides together:

`\begin{gathered} \text{Perimeter = a + b + c} \\ \text{Perimeter =}18\text{ + 24 + 25.16} \\ \text{Perimeter = 67.16 m} \end{gathered}`

To 3 significant figures, the perimeter is 67.2 m

b) To find the area since we have two sides and an angle, we will apply the formula:

`\text{Area of triangle = }(1)/(2)ab\sin C`
`\begin{gathered} \text{Area of the triangle = }(1)/(2)*\text{ 18}*24*\sin 72\degree \\ \text{Area of the triangle = 216}*0.9511 \\ \text{Area of the triangle = 205.4376 m}^2 \\ \\ To\text{ 3 significant figures, the area is }205m^2 \end{gathered}`