#2      54. MAC, JACK AN’ USSEN                             
Mac, Jack an’ ussen
Were feudin an’ cussin’
About this paper roll.
It fell out of the press
And at last came to rest
In this 100-centimetre hole.
As I try to describe
Its position, you jibe
“It looks like a Mexican hat!”
Well, excluding the crown,
Take the brim all aroun’:
What is the area of that?

 

 Hints and strategies 

Hint 1

Hint 2

Solution

Extensions
HINT 1

You will need some notation in order to try to calculate the areas of the circles. Try setting r and R for the radii. Now how are these related?
HINT 2

Can you put this diagram to good use?
SOLUTION

The area of the end of the roll is

A = R2 – r2

    = (R2 – r2)

    = 2500

    = 7854 cm2.
EXTENSION

This problem is curious because the result does not depend on the actual sizes of the circles – only their relative sizes.

1. Explore this by choosing a couple of values for R and determining what the values of r must be.

2. There are other problems with this minimal sort of information. Keep an eye out for them.