Alan

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Your answer is neat though Melody!  I didn't realise you could click on the circular icons in Desmos to switch the curves on and off. 

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Eliminate the y's:

2x² + 3x -1 = 5 - x²

Add x² - 5 to both sides

3x² + 3x - 6 = 0

Factorize

3(x+2)(x-1) = 0

so x = -2 and x = 1

when x = -2  y = 5 - (-2)² = 1

when x = 1  y = 5 - 1² = 4

So the intersections occur at (-2, 1) and (1, 4)

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You can construct the following two equations for the two unknowns:

x+y+1 = 3x -2y +14                              ...(1)

x+y+1+3x-2y+14+2x-y+4 = 29           ...(2)

Collect the x's and y's on one side of each equation and the constants on the other:

2x - 3y = -13                     ...(3)  (from (1))

6x -2y = 10                       ...(4)  (from (2))

Multiply (3) by 3:

6x - 9y = -39                    ...(5)

Subtract (5) term by term from (4)

7y = 49

Hence y = 7

Plug this back into (4)

6x - 2*7 = 10

6x = 24

x = 4

So the length of the equal sides is:  4 + 7 + 1 = 12 cm

Check:  3*4 - 2*7 +14 = 12 cm

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Multiply both sides by sinx

3tanx*sinx = 8

tanx = sinx/cosx so the above can be written as

3*sin²x/cosx = 8

Multiply both sides by cos x

3*sin²x = 8cosx

sin²x = 1 - cos²x so the above becomes

3*(1 - cos²x) = 8cosx

3 - 3cos²x = 8cosx

Add 3cos²x to both sides of the equation

3 = 3cos²x + 8cosx

Subtract 3 from both sides of the equation

0 = 3cos²x + 8cosx - 3

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I misread the question to say that pressure varies linearly with height.  Of course, it actually says the rate of change of pressure with height varies linearly with pressure. 

The question appeared twice, once in another language.  When I got a Google translation of that it was clear that it referred to the pressure being 0.83 atmospheres at 5000 metres.

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Actually these are representative of the approximate radii of the atoms.