Dr. Steve is working with a new radioactive substance in his lab. He currently has 64 grams of the substance and knows that it decays at a rate of 25% every hour.
Identify whether this situation represents exponential growth or decay. Then determine the rate of growth or decay, r, and identify the initial amount, A.
Remember that the general decay equation is: [tex]y=A(1-r)^{x}[/tex] where [tex]y[/tex] is the amount after a time [tex]x[/tex] [tex]A [/tex] is the initial amount [tex]r[/tex] is the the decay percent in decimal form
The first ting we are going to do is find [tex]r[/tex] by dividing our decay rate of 25% by 100%: [tex]r= \frac{25}{100} =0.25[/tex]. We also know from our problem that [tex]y=64[/tex]. Lets replace [tex]y[/tex] and [tex]r[/tex] in our formula: [tex]64=A(1-0.25)^{x} [/tex] [tex]64=A(0.75)^{x} [/tex] We know now that our decay rate is 0.75, and since 0.75<1, we can conclude that this situation represents exponential decay.
Now, to find the initial amount, we are going to solve our equation for [tex]A[/tex]: [tex]64=A(0.75)^{x} [/tex] [tex]A= \frac{64}{0.75 ^{x} } [/tex] Notice that [tex]A[/tex] will depend on the number of ours [tex]x[/tex].
