Alan

Here is my approach:

Yet another approach:

5*5^b = (5^2)^(b-1)

5^(b+1) = 5^(2*(b-1))

So b+1 = 2*(b-1)  or  b = 3

Let x = 0.4242424242....

Then 100x = 42.42424242....

Subtract x from 100x to get  99x = 42

Can you take it from here?

Maple produces the following:

0 = 0.972977645*x + 0.000012890360*x^2 - 0.24780681*10^(-8)*x^3 + 6.5559673*ln(x) - 4320.368041;  Using solve in Maple, x is                    

 4342.652916

Whatever temperature scale you are using it looks like the room will be pretty hot!!!
 

As follows:

I get the following:

For 1. see https://web2.0calc.com/questions/help-is-needed_7#r5

First consider the red dashed line.  The y-intercept is -1.  The slope is 2/1 = 2.

Therefore the equation of the line can be written as y = 2x - 1

The yellow region is below y, so the yellow region ca be expressed as  y < 2x - 1  

However, you are asked to exress it in the form ax + by + c >0, so subtract y from both sides of the previous equation to get 0 < 2x - y - 1

This is the same as  2x - y - 1 > 0

I've used > rather than >=, but if the yellow region is meant to include the red line then replace > by >=.

Answered here: https://web2.0calc.com/questions/expected-value_28#r1