# Suppose the total amount, A, of radioactive material present in the atmosphere at time T can be...

## Question:

Suppose the total amount, {eq}A {/eq}, of radioactive material present in the atmosphere at time {eq}T {/eq} can be modeled by the formula {eq}\displaystyle \int_0^T (Pe^{-r t})\ dt {/eq}, where {eq}P {/eq} is a constant and {eq}t {/eq} is time in years. Suppose that recent research suggests that {eq}r = 0.002 {/eq} and {eq}P {/eq} (present amount of radioactive material) {eq}= 200 {/eq} millirads. Estimate the total future buildup of radioactive material in the atmosphere if {eq}r {/eq} and {eq}P {/eq} were to remain constant.

## Integrating:

This problem will allow us to understand how to use an important result when integrating functions. The result that will be using is:

$$e^{-\infty}=\frac{1}{\infty}=\frac{\frac{1}{1}}{0}=0$$

See why we need to use this result below.

As we have been told to find the future total buildup, we assume that there is no upper limit and, therefore, we integrate from {eq}0 {/eq} to {eq}\infty {/eq} to find the total future buildup.

\begin{align} A&=\int_0^\infty 200e^{-0.002 t}\ dt\\ &=-\frac{200}{0.02}\left [ e^{-0.02t} \right ]_0^\infty\\ &\left ( \because e^{-\infty}=\frac{1}{\infty}=\frac{\frac{1}{1}}{0}=0 \right )\\ &=-10000\left [ 0-1 \right ]\\ &=10000 \end{align}