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Solve for X: Difficult (Algebra?)

645=(60/(.09(.91)^x)) + (1000/(1.09)^x)

How do I go about this? Can anyone get an answer or a way to get one? Calculator crashes...

MathsIsFun

Administrator

Registered: 2005-01-21

Posts: 7,711

Re: Solve for X: Difficult (Algebra?)

There may be no (real?) solution. The plot of the right hand side is: Plot of (60/(.09*(.91)^x))+(1000/(1.09)^x), which doesn't go down to 645. Unless I have mis-interpreted the formula.

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"The physicists defer only to mathematicians, and the mathematicians defer only to God ..."  - Leon M. Lederman

numen

Member

Registered: 2006-05-03

Posts: 115

Re: Solve for X: Difficult (Algebra?)

Is it still possible to get an expression for x?

This is where I get stuck:

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Bang postponed. Not big enough. Reboot.

Zhylliolom

Real Member

Registered: 2005-09-05

Posts: 412

Re: Solve for X: Difficult (Algebra?)

I don't believe it has a solution. I shall provide some graphical and analytical support for this claim.

To start, I began by attempting to simplify the expression in order to see if a solution could be seen that way. After a few steps, I arrived at the following:

I did not see a way that this could be solved, so I defined this expression as a function f:

Now the issue at hand is to see if f(x) is ever zero. Since algebraic techniques did not help in solving the problem, I graphed this function to see if it had a root. In fact, the lowest it seemed to go was approximately 600.78397, when x = 1.746827. Now the remaining problem is this: Is 600.78397 the only minimum value on the graph of our function f(x)?

To answer this, I will use calculus. I'm not sure if you're comfortable or not with this, but I will go ahead just to show that the problem has no solution. Now take the first derivative of f(x):

We know that the graphically observed point 1.746827 is a minimum, because df/dx | x = 1.746827 is zero. We have already graphically confirmed this, however. The real question is whether or not there are more minimums on f(x) which would possibly be low enough to allow a value of 0 for f(x). To investigate this we examine the second derivative:

Note that the second derivative is always positive, since the square of the natural logarithm is always positive, and a positive value taken to an exponent always results in a positive answer. Since the second derivative is always positive, then f(x) is concave up on (∞, ∞), which means its slope is always increasing. A result of this is that there will only be one minimum to f(x). But we have already found a minimum of f(x), as this can be the only minimum, there is no value for f(x) that is less that 600.78397, so it has no roots and thus the original equation posted has no solution.