Alan

Here's the answer to part 1:

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At the point in the rotation as it appears in the illustration, the horizontal force of the piston rods on the piston tend to want to push the piston to the right: i.e. towards the right side wall and away from the left side wall.  Hence there will be a non-zero reaction force from the right-side wall on the piston, but no reaction force from the left-side wall on the piston.  Later in the cycle, the opposite will hold.   

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As follows:

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If the domain of t lies between -infinity and +infinity, the range of the right hand side lies between 0 and 1.  Hence there is no value of t that allows the right hand side to equal 95350.

So you did. However

a = 6, b = 3, n = 27 results in 2105947230095214567

You still need to tighten your constraints!

a = 6, b = 3, n = 9 results in 61965

9 is not prime.

You’ll need tighter constraints than that!

With a = 4, b = 2, n = 8, the result is 8128.  n still not prime.

"(a^(n)b - ab^(n))/(2n)

When a and b are integers, and when the result is an integer too, i predict n is a prime number."

Hmmm!

a = 4, b = 2, n = 4 gives (aⁿb - abⁿ)/(2n) = 56

a, b and (aⁿb - abⁿ)/(2n) are all integers, but n isn't prime.

"Find projected population of location in 2015:

1. The population in Cedar Rapids, Iowa was 169775 in 1980 and 168767 in 1990.

 2. The population in Janesville, Wisconsis was 139420 in 1980 and 139510 in 1990."

You say nothing about the population growth model to be used!  I'll assume a simple exponential model:

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You need to know about a more advanced function called the LambertW function to solve this in terms of radicals etc.

The result, together with a numerical approximation, is as follows: