# Thread: Calculus word problem help.

1. ## Calculus word problem help.

At the end of a paper production line, paper is rolled onto an inner cylinder of
radius Rc. The paper can only be rolled up to a maximum radius Ro (Ro > Rc)
for storage reasons. The thickness of the paper (s) varies depending on the type
of paper, for instance tracing paper is thinner than photocopying paper, therefore
the total length of paper rolled up to Ro will also vary.

(a) Suppose that with paper of thickness s, a length of paper L results in a roll
of radius R (R = R(s;L)). Find an equation to model the change in radius of
the roll in terms of an increase in the length L given the thickness s of paper.

(b) Find the total length of paper in a cylinder of radius Rc = 3 cm if the radius
of the roll of paper is Ro = 7 cm if the paper has a thickness of 0.1 mm.

I have not got far this problem, just very stuck.

when the paper is rolled than the volume would be $\displaystyle V = \pi r^{2} W$ where 'w' is the width.

and when it is unrolled than the volume is $\displaystyle V = w .s .L$

w = width S = thickness, L = length

no idea what to do with the rest of the information? I think part 'a' is asking me to find $\displaystyle \frac{dR}{dL}$ ? Not sure how though?

Any help appreciated

2. come one Some one must know how to do this question (?) =) Any hint!!

3. Suppose the paper has width W. Then a length L of paper with width W and thickness s has volume LWs. A cylindrical roll of paper of width W and radius r will have volume $\displaystyle \pi r^2 W$. Setting those equal, $\displaystyle LWs= \pi r^2 W$ and W cancels out: $\displaystyle \pi r^2= Ls$.