WMA11 October 2022

8. WMA11/01 Edexcel IAL P1 October 2022, Q8 (Radians: Radian Measure & Trigonometry)

Figure 2 shows the plan view of a design for a pond.​​ 

The design consists of a sector AOBX of a​​ circle centre O joined to a quadrilateral AOBC.​​ 

• BC = 8m​​ 

• OA = OB = 3m​​ 

• angle AOB is​​ 2π3​​ radians​​ 

• angle BCA is​​ π6​​ radians​​ 

(a) Calculate​​ 

• the exact area of the sector AOBX,​​ 

• the exact perimeter of the sector AOBX.​​ 

(5)

(b) Calculate the exact area of the triangle AOB.​​ 

(2)

(c) Show that the length AB is​​ 33​​ m.​​ 

(2)​​ 

(d) Find the total surface area of the pond. Give your answer in m2​​ correct to 2 significant figures.​​ 

(5)

SOLUTION​​ 

i- Using the area of a sector formula.

 A=12r2θ

Where the angle AOBX is​​ 

θ=2π-2π3=4π3

So on substituting​​ 

A=12324π3

A=6π m2

ii-​​ 

The perimeter is the total external length of any shape. So in iur case, the perimeter would be equals to the sum of arc length and twice the radius. So let us first find the arc length.​​ 

l=rθ

l=3 x4π3

l=4π

Therefore, perimeter is​​ 

  P=l+2r

P=4π+6m 

b-​​ 

A=12a bsin⁡c

A=12x 32xsin⁡2π3=934m2

Area of ∆AOB=934m2

c- Using cosine rule to​​ find the length AB.

AB2=32+32-233 cos2π3

AB=32+32-233cos⁡2π3

AB=27

AB= 93

AB=33 m

d-​​ 

Total Surface Area=Area of Sector+Area of ∆AOB+Area of ∆ABC

Considering the triangle ABC, as shown below. Let us first find the angle y.​​ 

Using sine rule to find angle BAC.

sin⁡y8=sin⁡2633

y=sin-1⁡8sin⁡3633

y=0.87852…

Now, lets us find angle x as marked in triangle ABC

x=π-0.878 52+π6

x=1.7394….

Therefore, the area of traingle ABC.​​ 

Area of ∆ABC=12absin⁡C

Area of ∆ABC=12(AB)(BC)sin⁡x

Area of ∆ABC=12x 8 x 0.878 52+33x sin 1.7394

Area of ∆ABC=20.4896.. ..m2

Hence,​​ the total surface area is​​ 

Total Surface Area=Area of Sector+Area of ∆AOB+Area of ∆ABC

Total Surface Area=6π+934+20.4896

Total Surface Area=43.236 m2

Total Surface Area=43m22 sfg