Question

The amount of a radioactive material changes with time. The table below shows the amount of radioactive material f(t) left after time t:

t(hours) 0 1 2
f(t) 180 90 45

Which exponential function best represents the relationship between f(t) and t?

Answer 1

Answer:
`f(t)=180(0.5)^t`

Explanation:

The general exponential function is given by :-

`f(x)=Ab^t`, where A is the initial amount , b is the multiplicative rate of change and t is the time period.

From , the given table the value of the function at t=0 is 180.

i.e the initial amount of radioactive material = 180

Also, the multiplicative rate is the common ratio of the values of the function w.r.t. to time

The multiplicative rate =
`(90)/(180)=(1)/(2)=0.5`

Now, the exponential function best represents the relationship between f(t) and t will be :-

`f(t)=180(0.5)^t`

Answer 2

`\bf \begin{array}ccc t&0&1&2\\ f(t)&180&90&45 \end{array} \\\\[-0.35em] ~\dotfill\\\\ \qquad \textit{Amount for Exponential Decay}\qquad \stackrel{t}{0}~~,~~\stackrel{f(t)}{180}\\\\ A=I(1 - r)^t\qquad \begin{cases} A=\textit{accumulated amount}\to &180\\ I=\textit{initial amount}\\ r=rate\to r\%\to (r)/(100)\\ t=\textit{elapsed time}\to &0\\ \end{cases} \\\\\\ 180=I(1-r)^0\implies 180=I(1)\implies 180=I \\\\\\ \boxed{A=180(1-r)^t} \\\\[-0.35em] ~\dotfill`

`\bf \stackrel{t}{1}~~,~~\stackrel{f(t)}{90}\qquad \qquad 90=180(1-r)^1\implies \cfrac{90}{180}=1-r \\\\\\ \cfrac{1}{2}=1-r\implies r=1-\cfrac{1}{2}\implies r=\cfrac{1}{2} \\\\\\ A=180\left( 1-(1)/(2) \right)^t\implies A=180\left( \cfrac{1}{2} \right)^t`