IIT JEE , AIEEE Forum - Mathematics , Differential Calculus

ramkumar_november

A conical vessel is to be prepared out of a circular sheet of gold of radius r . How much sectorial area is to be removed from the sheet so that the vessel has maximum volume ?

(This question came in my preboard for 6 marks)........

please post the complete solution...........

ramkumar_november

experts please help..............

rockey

u have to assume the sectorial area as x then write total vol. in terms of it .
let that fununction be f(x)
max. that function to get required ans

ramkumar_november

rockey......... please post a detailed solution........

how will you write the total volume in terms of x????

joyfrancis

If the cone is made up of the sectorial area of the circle;
area of sector = curved surface area of cone

=> Lr/2 = Rl...(1)

Here L=length of arc of the sector cut out , r is the radius of the big circular disc, R is the radius of the base of the cone formed.

Now,
L = 2R....(2)

From equations 1 and 2 we have
r = l.......(3)
Volume of cone = 1/3((R²h))
& l² = R²+h²
.: V = 1/3(h(l²-h²))

But l=r
.: V = h(r²-h²)/3

Now we have V as a function of h.
f(h) = h(r²-h²)/3
=> f'(h) = (r²-3h²)/3

Crtical points are given by f'(h)=0
i.e (r²-3h²)/3 = 0
=> r²=3h²
=> r = hroot3....(neglect negative value)
This has to be maxima since minimum vol. is zero only.

Area of sector = Area of Cone = Rl
=Rr
=r*(root(l²-h²))
=r*(root(r²-r²/3))
=r²(root(2/3)).....(ans)

ramkumar_november

@joyfrancis...... i appreciate you for the efforts taken by you...... but you have misunderstood the question........ the cone is not made from the removed sectorial area .... it is made from the part left after removing the sectorial area..........

akhil_o

let height of cone=h
radius of cone=R

V=1/3 pi R²h
now h²=r²-R²
so
V=pi/3 R²r²-R^{2

differentiating and equating to 0}2r²=3R²
R=r

2/3

so the base circumference is
2 pi R
let the angle of cutt sector =A
(2pi-A)r=2 pi r

2/3

from this we get angle A=2 pi(1-

2/3)
so area of sector cut out
=(1-

2/3)r²

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