Sticky Tape
Work out the radius of a roll of adhesive tape.
Age
14 to 16
Challenge level
A roll of adhesive tape is wound round a central cylindrical core of radius 3cm.
The outer radius of a roll containing 20m of tape is 4cm.
What is the approximate outer radius of a roll containing 80m of tape?
The cross section of the $20 \; \text{m}$ tape has area:
$4^2 \pi- 3^2 \pi = 7\pi \text{cm}^2$
Therefore, the $80 \; \text{m}$ tape should have a cross-section area:
$4 \times 7\pi = 28\pi \; \text{cm}^2$
If $r$ is the outer radius of the $80 \;\text{m}$ roll,
$r^2 \pi- 3^2 \pi = 28\pi \text{cm}^2$
$\pi(r^2- 3^2) = 28\pi \text{cm}^2$
$r^2- 3^2 = 28 \text{cm}^2$
$r^2 = 37 \text{cm}^2$
Hence, the outer radius of the $80 \;\text{m}$ roll will be approximately $\sqrt{37} \;\text{cm}$