simplify x + 40 = (x/2)(e^-200/x) + (x/2)(e^200/x), and solve

nashbaker

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Oct 16, 2017
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All,

I cannot figure out how to show all steps to the following problem.

Original equation: x + 40 = (x/2)(e^-200/x) + (x/2)(e^200/x)

Solve for x.

Answer (derived from cad dimensions) : x = 506.53

Here is as far as I could simplify :

x + 40 = (x/2)(e^(-200/x)) + (x/2)(e^(200/x))
 
You cannot "solve" this one, either: \(\displaystyle x+2 = e^{x}\). There are Real Numbers solutions.

Who told you to "Solve for x" and why do you believe it is possible?
 
Original equation: x + 40 = (x/2)(e^-200/x) + (x/2)(e^200/x)

Here is as far as I could simplify :
x + 40 = (x/2)(e^(-200/x)) + (x/2)(e^(200/x))

Presumably it is the second of these that represents what you intended (though you didn't simplify anything); Denis took the first one, which means

. . . . .\(\displaystyle x\, +\, 40\, =\, \left(\dfrac{x}{2}\right)\, \left(\dfrac{e^{-200}}{x}\right)\, +\, \left(\dfrac{x}{2}\right)\, \left(\dfrac{e^{200}}{x}\right) \)

and which could be solved algebraically (but is too trivial to be what you meant); the second is

. . . . .\(\displaystyle x\, +\, 40\, =\, \left(\dfrac{x}{2}\right)\, \left(e^{\frac{-200}{x}}\right)\, +\, \left(\dfrac{x}{2}\right)\, \left(e^{\frac{200}{x}}\right) \)

and can only be solved by numerical approximation methods (which a CAD program will use). Wolfram Alpha gives the solution you provided, for this version.
 
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