Question

Consider the exponential model g(x) = 96(69/96)^(x/5). Find the continuous decay rate.

Answer 1

The continuous decay rate is 0.066.

Explanation:

The continuous growth/decay rate is typically represented by the constant term k in the exponential formula:

`\large\boxed{f(x)=Ae^(kx)}`

Given exponential model:

`g(x)=96\left((69)/(96)\right)^{(x)/(5)}`

Rewrite both equations by applying the power of a power exponent rule:

`f(x)=A\left(e^(k)\right)^x`

`g(x)=96\left(\left((69)/(96)\right)^{(1)/(5)}\right)^x`

Now, we can find the value of k by equating the relevant parts of the two functions:

`e^(k)=\left((69)/(96)\right)^{(1)/(5)}`

Take natural logs (ln) of both sides of the equation:

`\ln\left(e^(k)\right)=\ln \left(\left((69)/(96)\right)^{(1)/(5)}\right)`

Apply the power log rule:

`k\ln\left(e\right)=(1)/(5)\ln \left((69)/(96)\right)`

Given ln(e) = 1:

`k=(1)/(5)\ln \left((69)/(96)\right)`

Computing the value of k gives:

`k=-0.0660483373...`

Therefore, as k is negative, this indicates a continuous decay rate of 0.066.