Question

Q(t)=Q(0)e^{pt} The above model describes the exponential decay of chemical element. t is the time in years, Q(0) is the initial amount of the chemical element, Q(t) is the amount of chemical element after t years and p is a constant. It is known that every 1600 years the amount of the chemical element drops to half ( its half-life is 1600 years). If the term e^p in the above equation can be replaced by the term x^y, where both x,y rational numbers, what is x and y?

Answer 1

If the term e^p is replaced by x^y where x and y are rational numbers, then the value of x and y are as follows:

x = 1/2

y = 1/1600

Step-by-step explanation:

Given

the exponential decay equation

$Q(t)=Q(0)e^{pt}$

Where $Q(t)$ is amount after $t$ years

$Q(0)$ is the initial amount

The half-life of the chemical element is 1600 years

Thus, in 1600 years

$Q(t)=Q(0)/2$

Therefore,

$\frac{Q(0)}{2}=Q(0)e^{p\times 1600}$

$\implies e^{1600p}=\frac{1}{2}$

$\implies (e^p)^{1600}=\frac{1}{2}$

$\implies (e^p)=(\frac{1}{2})^{1/1600}$

Now if we take $x=\frac{1}{2}$ and $y=\frac{1}{1600}$

Then $e^p$ can be replaced by $x^y$

Here x and y are rational numbers.

Hope this answer is helpful.

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