Question #197963

A circular cone piece of cardboard with a diameter of 1 m will be made into a conical hat 44 cm high by cutting a sector off and joining the edges to form a cone. Determine the anglein degree subtended by the sector removed.


1
Expert's answer
2021-05-25T10:19:14-0400
r=100 cm/2=50 cm,h=44 cmr=100\ cm/2=50\ cm, h=44\ cm


From the right triangle


x=r2h2=502442=2141(cm)x=\sqrt{r^2-h^2}=\sqrt{50^2-44^2}=2\sqrt{141}(cm)

Let:

C1=C_1= circumference of the circle


C1=2πrC_1=2\pi r

C2=C_2= circumference of the base of the cone


C2=2πxC_2=2\pi x

C=C= length of the arc


C=rθC=r\theta



Then


C=C1C2C=C_1-C_2

rθ=2πr2πxr\theta=2\pi r-2\pi x

θ=2π(rx)r\theta=\dfrac{2\pi(r-x)}{r}

θ=2π(1xr)\theta=2\pi(1-\dfrac{x}{r})

θ°=2π(1xr)(360°2π)\theta\degree=2\pi(1-\dfrac{x}{r})(\dfrac{360\degree}{2\pi})

θ°=(1xr)360°\theta\degree=(1-\dfrac{x}{r})360\degree

θ°=(1214150)360°189°\theta\degree=(1-\dfrac{2\sqrt{141}}{50})360\degree\approx189\degree


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