WMA11 Jan 2022

5. WMA11/01 (Edexcel) IAL P1 January 2022 Q5 Radian Measure, Area &​​ Perimeter of Segments​​ 

Figure 2 shows a plan view of a semicircular garden ABCDEOA​​ 

The semicircle has​​ 

• centre O​​ 

• diameter AOE​​ 

• radius 3m​​ 

The straight line BD is parallel to AE and angle BOA is 0.7 radians.​​ 

(a) Show that, to 4 significant figures, angle BOD is 1.742 radians.​​ 

(1)

The flowerbed R, shown shaded in Figure 2, is bounded by BD and the arc BCD.​​ 

(b) Find the area of the flowerbed, giving your answer in square metres to one decimal place.​​ 

(3)

(c) Find the perimeter of the flowerbed, giving​​ your answer in metres to one decimal place.​​ 

(3)

SOLUTION​​ 

a-

OB^D=AO^B=0.7rad alternate angles

Whereas, ∆BOD is an isoscles triangle.​​ 

BD^O= OB^D base angles of isosceles ∆

Since a straight line angle is equals ​​ to π. So,​​ 

BO^D=π-0.7+0.7

BO^D=1.74159…

∡BO^D=1.742

b-​​ To find the area of the flowerbed, subtract from the area of a sector, the area of an isoscles​​ triangle.​​ 

Area of R=Area of sector-Area of isoscles triangle

Area of R=12r2θ-12r2sin⁡θ

Area of R=12r2θ-sin⁡θ

Area of R=12321.742-sin⁡1.742

Area of R=3.4026 m2

Area of R=3.4m2

c-​​ To find the perimeter of the flowerbed,add the arc length BCD and length BD.​​ 

Perimeter=LBCD+BD 

Where,​​ 

Arc length is given as l=rθ.

LBCD=3 x 1.742

LBCD=5.226m

And, length BD is found by applying sine rule to the triangle OBD.​​ 

BDsin 1.742=3sin⁡0.7

BD=3sin⁡1.742sin⁡0.7

BD=4.59m

Putting the respective values in the perimeter expression.​​ 

P=3(1.742)+3sin⁡1.742sin⁡0.7

P=5.226+4.59

P=9.8147

P=9.8m