Borbin the 🐱

The final one:

Using trigonometric identities, the expressions are divided into sin(x)/x segments for which the lim exists. Rewriting the expression to remove the denominator is one way of doing it, but won't work in such cases. Using a known lim makes it all so easy!

The third one:

It is c) that gets to the solution! The expression f(x) + f(y) + x^2y + xy^2 is part of the binominal (x+y)^3.
If f(x) is the first/last element of the binominal, the equation can be satisfied!

Here is second one:

The trick is to bring the x to the right side, and set x to 0. This is the first equation for b.
Then with b in place, get the second equation. Isolate a and substitute 0 for x. This gets a

I'm helping out with Highschool math, and some of the questions are so cool!

Here is one:

The solution is to get the two equations from the derivative. With a 4-th order polynomial we get 3 points of equal slope, but only 2 have a common Tangent line.
Then it gets super easy!

The custom airfilter box needed an update. Too much air flow caused a lean mixture.
The solution is an adjustable air flow.

The tubes were made out of stainless steel, grade 304.

Changed the filter from 6" length to 5" length (K&N Filter RD-0610) to cover for the air adjustment lever.

The current airfilter box of my motorcycle is undersized. Way too small and has the wrong connection tubes to the carbs which are not symmetrical. Note the diameter getting larger towards the carbs!
The restricted airflow makes it also very difficult to dial in the correct fuel-air-ratio.

To change this, I designed a new airfilter box that takes care of all the shortcommings:

Here are some pictures of the 'making of':