Part 1 of 3 Points: 0 of 1 Although primarily used for relief of pain, opioids have addictive properties that often make withdrawal difficult. Deaths through unintentional drug overdose have risen sharply across a certain country since the early 2000 s. A particular state has an exceptionally high rate of drug overdose death. The number of deaths in the state in year $t$, where $t$ is the number of years after 2004 , can be approximated by $D(t)=326(1.07)^{t}$. a) How many unintentional drug overdose deaths were there in the state in 2014 ? b) In what year were there about 2,500 unintentional drug overdose deaths in the state? c) What is the doubling time of unintentional drug overdose deaths in the state?
Solution
Step 1 ：The problem provides the function \(D(t)=326(1.07)^{t}\), where \(t\) is the number of years after 2004, and \(D(t)\) represents the number of unintentional drug overdose deaths in the state in year \(t\).
Step 2 ：For part a, we are asked to find the number of unintentional drug overdose deaths in the state in 2014. Since 2014 is 10 years after 2004, we substitute \(t=10\) into the function.
Step 3 ：By substituting \(t=10\) into the function, we get \(D(10)=326(1.07)^{10}\).
Step 4 ：Calculating the above expression, we find that the number of unintentional drug overdose deaths in the state in 2014 is approximately 681.
Step 5 ：Final Answer: The number of unintentional drug overdose deaths in the state in 2014 is approximately \(\boxed{681}\).
