Give an exact expression for the doubling time of the function y = 80 (1.4)^x/4, and quote an approximate numerical value.

Question

Give an exact expression for the doubling time of the function y = 80 (1.4)^x/4, and quote an approximate numerical value.

what I did: y = 80 (1.4)^x/4

         y = 80 (1.4)^2/4
         y = 80 (1.4)^1/2
         y=56

is that the correct?

Answer 1

A naive way to do it is:

1. plug-in a value for $x$
2. compute for the corresponding $y$.
3. plug-in double the computed value of $y$
4. solve for the corresponding $x$ value for the doubled $y$
5. take the positive difference of the the solved value of $x$ and the first plugged-in value for $x$

Say for your question:

1. Plug-in $x=0$ into $y=80(1.4)^{\frac{x}{4}}$
2. We obtain $y=80$ since $80(1.4)^{\frac{0}{4}}=80$
3. Plug-in $y=160$ into $y=80(1.4)^{\frac{x}{4}}$
4. Solving for $x$ in $160=80(1.4)^{\frac{x}{4}}$ we obtain $x= 4 \log_{1.4} 2$
5. Getting the difference of the two $x$-values, we have the doubling time of $4 \log_{1.4} 2$

Answer 2

The doubling time is the difference in $x$ that doubles the value of $y$. Pick a starting $y$. You might just plug in $x=0$ and compute $y$. Now find the $x$ which corresponds to $2y$. The difference in the $x$'s is the doubling time.