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# Geometry question!

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Johnny wants to find the volume of the paper in a fresh roll of paper towels.

Please explain what he should do to calculate the volume of the paper. Be sure to include any equipment/formulas he'll need.

Feb 12, 2019

#1
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$$\text{he'll need the radius of the roll }R\\ \text{the radius of the inner cardboard tube }r\\ \text{and the height of the roll }h\\ V = \pi h (R^2 - r^2)$$

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Feb 12, 2019
#2
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Imagine this.

You have a big solid paper cylinder.   The volume is the area of the base* height    =      $$\pi*Radius^2*height$$

Now you drill out the centre of it.  This is a smaller cylider, the hollow bit in the middle.

The volume of the the small drilled out cylinder is     also     $$\pi*\color{red}{r} \color{black}adius^2*height$$      I used a little r this time becaus it is a smaller radius.

Anyway I don't want the middle bit, so I throw it away.

The volume of paper is

$$V=\pi R^2h \quad- \quad \pi\color{red}{r} \color{black}^2h\\$$

This is the same as Rom's answer - his answer is just factorised.

Feb 12, 2019