Find the exponential decay model that satisfies , if the halving time is years. Round any numerical quantities to 6 decimal places if need be.

Question

Find the exponential decay model  that satisfies  , if the halving time is  years. Round any numerical quantities to 6 decimal places if need be.

Quizplus · Accepted Answer

The general form of an exponential decay model is $y(t)=a\cdot e^{-kt}$, where $y$ is the amount (or value) at time $t$, $a$ is the initial amount (or value), and $k$ is the decay constant.

If the halving time is $h$ years, then we know that $a\cdot e^{-kh}=0.5a$, which implies that $e^{-kh}=0.5$.

Taking the natural logarithm of both sides, we get:

$$\ln(e^{-kh})=\ln(0.5)$$

$$-kh=\ln(0.5)$$

$$k=-\frac{\ln(0.5)}{h}$$

Therefore, the exponential decay model that satisfies the given condition is:

$$y(t)=a\cdot e^{-\frac{\ln(0.5)}{h}t}$$

Substituting the given value of $h=30$ years, we get:

$$y(t)=a\cdot e^{-\frac{\ln(0.5)}{30}t}$$

This matches the form of choice (C) with the same decay constant.

Find the Exponential Decay Model That Satisfies