Example: The half life of radium is 1690 years. If 50 grams are present now, how much will be present in 630 years?
Solution: There is a two part process to this problem. Part 1: Use some of the information to find the decay rate of radium. Part 2: Answer the question using the rest of the given information.
Part 1: Find the decay rate of radium.
Since we are using an exponential model for this problem we should be clear on the parts of the exponential decay model.
Exponential Decay Model
is the initial amount
is the decay rate
is the time
is the amount after t time has passed
Since radium has a half life of 1690 years, we know that whatever initial amount of radium is present after 1690 years there will be half of that initial amount left. This allows me to identify when the .
Substitute these values into the exponential decay formula and solve for k.
Using only the information about radium having a half-life of 1690 years I have found the decay rate for radium.
Note: Although I have put an approximation for k here, try not to round until the very last step.
Part 2: Answer the question using the rest of the given information.
Given information: If 50 grams are present now, how much will be present in 630 years?
With this information I can identify the initial amount of radium as 50 grams and the time to be 630 years. Symbolically that is when the . Substitute the given information and the decay rate k found in part 1 to the exponential decay formula.
, , and
38.615 grams will be present 630 years later is 50 grams are present initially.
What is the interest rate necessary for an investment to quadruple after 7 year of continuous compound interest?
Since this question involve continuous compound interest, we will use the associated formula.
We are given that the invest quadruples in 7 years. This tells me that when t=7 that A will be 4 times P. I can write that in symbols A=4P.
Substitute these values into the continuous compound formula and solve for the interest rate.
|Substitute the values of t and A into the formula|
|Solve for r by dividing both sides by P and simplifying|
|Solve for r by taking the log of both sides.|
|Solve for r by dividing both sides by 7 and simplifying|
|Find the value in the calculator|
|Write the answer as a percentage rounded to two decimal places|