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Weekly Challenge 43: A Close Match

Can you massage the parameters of these curves to make them match as closely as possible?

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Weekly Challenge 44: Prime Counter

A weekly challenge concerning prime numbers.

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Weekly Challenge 28: the Right Volume

Can you rotate a curve to make a volume of 1?

Weekly Challenge 23: Stand up Arcs

Stage: 4 and 5 Short Challenge Level: Challenge Level:1

The triangle made up of the stand up arcs has area $\frac{1}{2}\times r \times \frac{1}{2}\pi r$ which is $\frac{1}{4} \pi r^2$. This is equal to the area of a quarter of the unit disc so the unit disc has area $\pi r^2$.
 
Suppose you straightened complete circuar arcs.Then the tops would lie on the line $y = 2 \pi x$ and the area of the triangle made from these stand up arcs would be $\frac{1}{2}\times r \times2 \pi r$ which is $\pi r^2$. This gives the area of the whole unit disc.