SOLUTION: A certain type of decorative tissue paper is sold in rolls of radius 5 cm with a cardboard core of radius 1 cm. If the paper is used until the thickness of the paper layer is reduc

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Question 257926: A certain type of decorative tissue paper is sold in rolls of radius 5 cm with a cardboard core of radius 1 cm. If the paper is used until the thickness of the paper layer is reduced to half its original thickness, what fraction of the original amount of tissue paper is still on the roll?
Answer by Theo(13342) About Me  (Show Source):
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cross section area of the full roll is equal to:

pi * r^2 = pi * 5^2 = pi * 25.

this is not all tissue however.

the core of the roll is empty so we have to subtract that.

cross section area of the core of the roll is equal to:

pi * r^2 = pi * 1^2 = pi.

if we subtract pi from 25 * pi we get 24 * pi which is the cross section area of the tissue on the roll when it is full.

the thickness of the paper layer is equal to 4 cm

half the thickness of the paper layer is therefore equal to 2 cm.

the radius of half the thickness of the paper layer on the roll is therefore 2 + 1 = 3 cm.

the cross section area of the half a roll of tissue paper is equal to:

pi*r^2 = pi*3^2 = pi*9

we have to subtract the core area from this which we know is pi, so we get:

9*pi = pi = 8*pi.

cross section area of a full roll is 24*pi.

cross section area of a half roll is 8*pi.

fraction of the original amount of paper on the roll is equal to 8*pi / 24*pi = 1/3