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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.

Answer and Explanation:

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.

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


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Integration Problems in Calculus: Solutions & Examples

from AP Calculus AB & BC: Homework Help Resource

Chapter 13 / Lesson 13
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