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In a circular sheet of paper of radius 15 cm a sector of 144° is removed and remaining is used to form a conical surface then the angle at the vertex will be?

In a circular sheet of paper of radius 15 cm a sector of 144° is removed and remaining is used to form a conical surface then the angle at the vertex will be?

Grade:9

1 Answers

Ram Kushwah
110 Points
2 years ago
Let the radius of base of cone formed is R ,slant height is l and height is h
 
When a arc is cut and formed into a cone as shown in the figure above then
The Slant height l= Radius of arc=15 cm
and Circumference of the base 2πR = Arc length
 
Calculation of arc length
 
In our question radius of circle r =15 cm
 
From the circle 144° sector is remove
angle of remaing arc=360°-144°=216°
Thus arc length= θ/360 x 2πr
=216/360 x 2π x 15
=18π
Thus 2πR=18π
 
R=9 cm
In the Δ AOB
 
sinθ=OB/AB=9/15=3/5
 
θ= sin⁻¹ (3/5)
 
Thus the angloe of vertax
∠ A =2θ = 2sin⁻¹ (3/5)

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