spiral problem: roll of tape

Question

i worked in cabinetry, and i tried a bunch of times to apply calculus to the roll of edge tape. the final diameter is the roll the tape is on, the beginning diameter is the roll plus the tape. the tape has a thickness. i would like to know the length of the tape and the rate of change of the diameter. i just went back to it and i'm still not quite sure. i have completed calc III and have still not touched this.

Answer 1

Each time the tape wraps around, how much does the radius, and therefore the circumference, increase?

Answer 2

While calculus is certainly an option for this question, I would like to point out that it's not necessary. Consider this: Your roll of tape has some volume. You can find this pretty easily with the volume formula for cylinders: $\pi hr^2$, or for this case $\pi wr^2$ where $w$ is the width of the tape. Now clearly, if you unwind the entire roll of tape, the volume won't change. So, we can also write down an expression for the volume of the tape when unwound: $wlt$ where $l$ is the lenght, and $t$ the thickness. Setting these equal to each other and solving: $$\begin{aligned} wlt&=\pi wr^2 \\ lt&=\pi r^2 \\ l&=\frac{\pi r^2}t \end{aligned}$$ we get an expression for the length of tape. No calculus needed.

Answer 3

Thickness tape is p

N is the number of turns.

Initial radius + p *n is the total radius

Each wind has the circumference pi*(r+p*n)^2 The sum from 1 to n of the above equation is your length.