# Question d14f7

Mar 29, 2017

$7582$ sheets of paper

#### Explanation:

Once you have all the information, we have to decide how to use it.

Do not get confused by regarding the paper on the roll as a spiral - the radius keeps changing and the length of paper on each turn also gets longer.

the clue is in the first question which asks for the volume of paper on the roll.

Volume of a cylinder: $V = \pi {r}^{2} h$

However, one part of the volume of this roll is made of paper and another part is the volume of the wooden bit in the the centre.

d)i) Volume paper = Whole volume - volume of wood

The whole cylinder has its radius $= 30 c m$

Wooden cylinder has its radius $= 2 c m$

They both have a height of 21cm.

V_("paper") = color(red)(V_("whole cylinder")) - color(blue)(V_("wood")) 

${V}_{\text{paper}} = \textcolor{red}{\pi {\left(30\right)}^{2} \times 21} - \textcolor{b l u e}{\pi {\left(2\right)}^{2} \times 21}$

${V}_{\text{paper}} = \textcolor{red}{18 , 900 \pi} - \textcolor{b l u e}{84 \pi} = 18 , 816 \times \pi$

V_("paper") = 59,112 cm^3" "larr# this is the first answer

$0.125 m m = 0.0125 c m \text{ } \leftarrow$ this is the second answer.

But what is the length of the paper wrapped around to form the cylinder?

You have to imagine the whole length of paper being unrolled into one very long strip of paper.

It will form a cuboid, (a rectangular prism) with a length, a width and a height - ( the thickness of one sheet of paper).

The length we do not know, but the width is $21 c m$ and the thickness is $0.0125$ cm)

The VOLUME of the paper is the SAME! - whether it is rolled into a cylinder, or rolled out along the ground. We know the Volume!

So, now we have:

$V = l \times b \times h = 59 , 112 c {m}^{3}$

$l \times 21 c m \times 0.0125 c m = 59 , 112 c {m}^{3}$

$l = \frac{59112}{21 \times 0.0125} c m$

$l = 225 , 189 c m \text{ } \rightarrow$ this is a length of $2.25 k m$ of paper

It is cut into strips which are $29.7 c m$ long.
(These are the normal measurements of A4 paper)

$225 , 189 \div 29.7 = 7582$ sheets of paper