This post contains the answers to this week's Sunday Afternoon Maths and some extension problems based around the originals.

#### Two Lines

Let A have the equation

Therefore,

Which rearranges to

So

Substituting back in, we find

The co-ordinates of the point of intersection are

*y*=*mx*+*c*. B will have the equation*y*=*cx*+*m*.Therefore,

*mx*+*c*=*cx*+*m*.Which rearranges to

*x*(*m*-*c*) =*m*-*c*.So

*x*= 1.Substituting back in, we find

*y*=*m*+*c*.The co-ordinates of the point of intersection are

**(1,**.*m*+*c*)##### Extension

Let

and

*a*,*b*and*c*be three distinct numbers. What can you say about the points of intersection of the parabolas:*y*=*ax*^{2}+*bx*+*c*,*y*=*bx*^{2}+*cx*+*a*,and

*y*=*cx*^{2}+*ax*+*b*?#### Odd Sums

They are all equal to one third.

The sum of the first

*n*odd numbers is*n*^{2}(this can be proved by induction). This means that (sum of the first*n*odd numbers) ÷ (sum of the next*n*odd numbers) is^{n2}/_{(2n)2-n2}=^{n2}/_{4n2-n2}=^{n2}/_{3n2}=^{1}/_{3}##### Extension

What is (sum of the first

*n*odd numbers) ÷ (sum of the first*n*even numbers)?*x*^{xxxxx...} Again

*y*=

*x*

^{xxxxx...}so

*y*=

*x*

^{y}= e

^{y ln x}.

By the chain rule and the product rule,

^{dx}/

_{dy}= e

^{y ln x}(

^{dx}/

_{dy}ln

*x*+

^{y}/

_{x}).

Rearranging, we get

^{dx}/

_{dy}=

^{yey ln x}/

_{x(1-ey ln x ln x)}.

This simplifies to

^{dx}/

_{dy}=

^{xxxxxx...xxxxxx...}/_{x(1-xxxxxx... ln x)}##### Extension

What would the graph of

*y*=*x*^{xxxxx...}look like?#### Folding Tube Maps

Once the map is folded, it will look like this:

For the final tetrahedron to be regular, the red lengths must be equal. Let each red length be 2 (this will get rid of halves in the upcoming calculations). By drawing a vertical line in we can work out the width and height of the rectangle:

The width of the rectangle is 3 (one and a half red lengths). Using Pythagoras' Theorem in the blue triangle, we find that the height of the rectangle is √3. Therefore, the ratio of the rectangle is √3:3 or

**1:√3**.##### Extension

If the ratio of the rectangle is 1:

*a*, what is the ratio of the lengths of the sides of the tetrahedron?
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