WMA11 Jan 2023

• WMA11/01 (Edexcel) IAL P1 January 2023, Q6 (Radian: Radian Measure & Trigonometry)

Figure 1 shows the plan view for the design of a stage.

The design consists of a sector OBC of a circle, with centre O, joined to two congruent triangles OAB and ODC.

Given that

• angle BOC 2.4 radians

• area of sector BOC = 40m²

• AOD is a straight line of length 12.5m

• find the radius of the sector, giving your answer, in m, to 2 decimal places,

(2)

• find the size of angle AOB, in radians, to 2 decimal places.

(1)

Hence find

• the total area of the stage, giving your answer, in m³, to one decimal place,

(3)

• the total perimeter of the stage, giving your answer, in m, to one decimal place.

(4)

SOLUTION

a-​​ 

Using the formula of area of a sector.

A=12r2θ

40=12r22.4

40=1.2r2

r=401.2=1033 

r=5.7735

Radius is required in 2 decimal places.

r=5.77 m

b- ​​ Since​​ triangle AOB & COD are same triangles because their corresponding sides are equaivalent as OA=OD=6.25 & OB=OC= radius. Therefore, the sum of angle AOB, COD and BOC is​​ π or 180°.  ​​​​ 

AO^B= CO^D=θ

2θ+2.4=π

θ=π-2.42

θ=0.37079

θ=0.37 rad

c- The total area of the stage is equals to​​ 

Area of Stage=Area of Sector BOC+Area ∆AOB

Where​​ Area ∆AOB=Area ∆BOC

Area of Stage=40m2+2 x Area ∆AOB

Area of Stage=40+2 12 x 6.25  x1033 xsin⁡0.3707…

Area of Stage=53.0754

Area of Stage=53.1 m2

d-​​ The​​ total perimeter of the stage is

Perimeter=AO+AB+CD +arc length BC

AB=6.252+10332-26.251033cos⁡0.3707…

AB=2.2653….. 

Where,​​ CD=AB=2.2653

Now, using the arc length formula to find arc length BC.

l=rθ

l=1033 x 2.4 

l=83

Hence, the perimeter is​​ 

P=AO+AB+CD +arc length BC

P=12.5+22.2653+83

P=30.887

P=30.9 m