Why do I have to double an exponential growth and half an exponential decay function to find out how fast/slow the process is?Let’s take the exponential growth function to be y = A 0 e k t and exponential decay function to be y = A 0 e − k t .I’m just told that to find out How fast the process is, you have to double the exponential growth function and Half-time the exponential decay function. Why is this so? I never seem to understand it. Is there a way to prove it? Or is my tutor right by just telling me the fact and nothing else?

Question

Why do I have to double an exponential growth and half an exponential decay function to find out how fast/slow the process is?Let’s take the exponential growth function to be   y  =      A    0        e          k      t       and exponential decay function to be   y  =      A    0        e          −      k      t      .I’m just told that to find out How fast the process is, you have to double the exponential growth function and Half-time the exponential decay function. Why is this so? I never seem to understand it. Is there a way to prove it? Or is my tutor right by just telling me the fact and nothing else?

Sugainkohr · Accepted Answer

Let&#039;s take the exponential growth function   f  (  t  )  =      A    0        e          k      t      . It is simple to find       A    0   - this is just the value of f at time 0. But how can we find the value of k ?Suppose f(t) has value       y    0   at time       t    0   and value   2      y    0   at time       t    1  . Then  2      y    0    =  2      A    0        e          k              t        0              =      A    0        e          k              t        1              ⇒  2      e          k              t        0              =      e          k              t        1              ⇒  ln  ⁡  (  2  )  +  k      t    0    =  k      t    1    ⇒  k  =            ln      ⁡      (      2      )                      t        1            −              t        0            The time interval       t    1    −      t    0   during which f(t) doubles is called the &quot;doubling time&quot;. Notice that it does not depend on when it is measured. Once you know the doubling time you can find the growth constant k.You can do the same calculation with exponential decay except you measure the time taken for f(t) to halve, not double, which is called the &quot;half life&quot;.

Answered question

Answer & Explanation

New Questions in Algebra I