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? f(t) = 0.5(180)t f(t) = 180(0.25)t f(t) = 180(0.5)t f(t) = 0.5(50)t

Answer:- The exponential function best represents the relationship between f(t) and t is [tex]f(t)=180(0.5)^t[/tex] Explanation:-The exponential function is given by [tex]f(t)=Ab^t[/tex] , where A is the initial amount , b is the rate of change and t be the time period. According to the given table , at t=0 hour the exponential function =f(0)=180 , therefore A =180 At t=1 hour ,the exponential function [tex]=f(1)=180b^1\\\Rightarrow\ 90=180b\\\Rightarrow\ b=\frac{90}{180}=\frac{1}{2}=0.5[/tex]Thus by substituting the value of A and b in the function we get the required exponential function =[tex]f(t)=180(0.5)^t[/tex] , where t is the time period.