The amount of a sample of a radioactive substance remaining after t years is given to be a function of the form $Q ( t ) = Q _ { 0 } e ^ { - k t }$ . If 100 grams remain after 1,000 years and 90 grams after 2,000 years, what is the initial amount $Q _ { 0 }$ ? Round your answer to two decimal places.
A) 110.21 grams
B) 110.83 grams
C) 111.11 grams
D) 111.49 grams

Question

Quizplus · Accepted Answer

Given the formula $Q(t) = Q_0 e^{-kt}$, we can substitute the given values to find $Q_0$. For 1000 years and 100 grams, we have $100 = Q_0 e^{-1000k}$. For 2000 years and 90 grams, we have $90 = Q_0 e^{-2000k}$. Dividing the second equation by the first gives us a ratio that allows us to solve for $k$, and then we can solve for $Q_0$. The calculations lead to $Q_0 = 111.11$ grams.

The Amount of a Sample of a Radioactive Substance Remaining Q(t)=Q0e−ktQ ( t ) = Q _ { 0 } e ^ { - k t }Q(t)=Q0​e−kt

The Amount of a Sample of a Radioactive Substance Remaining $Q ( t ) = Q _ { 0 } e ^ { - k t }$