Question

An elastic belt is placed around the rim of a pulley of radius 5 cm. (Fig. 10) From one point C on the belt, the elastic belt is pulled directly away from the centre 0 of the pulley until it is at P, 10 cm from the point 0. Find the length of the belt that is still in contact with the pulley. Also find the shaded area. (use $\pi =3.14$ and √3 =1.73)

Answer 1

Answer:

Length = 20.93 cm.

Shaded area is 17.13 cm²

Step-by-step explanation:

Find Length OP:

OP = 10 cm

Find ∠AOP:

∠OAP = 90º (AP is tangent to the circle)

Cos θ = adj/hyp

cos (∠AOP) = 5/10

∠AOP = cos⁻¹ (5/10) = 60º

Find ∠AOB:

∠AOB = ∠AOP + ∠POB

∠AOB = 60 + 60 = 120º

Find ∠AOB (reflex):

∠AOB (reflex) = 360 - 120 = 240º

Find the length arc AB (Major):

Length of Arc AB (Major) = 240/360 x 2π(5) = 20.93 cm

Find the area of the of the sector ABC:

Area = 120/360 x π(5)² = 26.17  cm²

Find the triangle AOP:

Area = 1/2 ab sinC

Area = 1/2 (5) (10) sin (60) = 21.65 cm²

Find the area of ABP:

Area = 21.65 x 2 = 43.4 cm²

Area of the area of the shaded region:

Area = 43.4 - 26.17 = 17.13 cm²

Answer: The length of the belt in contact with the pulley = 20.93 cm. The shaded area is 17.13 cm²