I encountered a problem trying to find a solution to the following equation (the problem 9.45 from the wonderful textbook on Applied Calculus by Hoffman et.al (2013, p. 719)):

$$ A(t) = \frac{3}{k} (1 - e^{-kt}) $$

According to the exercise, we know that $ A(1) = 2.3 $ implying that

$$ 2.3 = \frac{3}{k} (1 - e^{-k}) $$

Here I don't know what to do. I played with the equation back and forth trying to multiply and divide both sides of it with all sorts of things but it did not bring me much. Of course, I can multiply both sides by $ k $

$$ 2.3 k = 3 (1 - e^{-k}) $$

And it looks like $ k = 0 $, which cannot be correct as the whole equation does not make sense in terms of the context the problem is given (it is about the content of a drug in patience's bloodstream, therefore there has to be a real-number solution.)

Wolfram Alpha suggests that the answer is $ k = 0.557214 $

Could you suggest me an analytical solution to the equation?

  • $\begingroup$ This equation is a "transcendental equation", so there is likely no simple formula for $k$ in terms of so-called "elementary functions". But there are plenty of ways of approximating the solution, which is what Wolfram Alpha has done. $\endgroup$ – Morgan Sherman Mar 21 '17 at 22:26
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    $\begingroup$ I asked a similar question before and was given a helpful answer here math.stackexchange.com/questions/1735552/… $\endgroup$ – WaveX Mar 21 '17 at 22:28

The "analytical" answer is $$k = \frac{30}{23} + W\left(-\frac{30}{23} e^{-30/23}\right) $$ where $W$ is the Lambert W function.

But I doubt that Hoffman et al intended you to solve it this way.


Just a note: it cannot be $k=0$ not only because it does not make sense, but because you would be dividing by 0 in the first equation.


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