# Algebra II : Radioactive Decay Equations

## Example Questions

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### Example Question #1 : Radioactive Decay Equations

Over the past few years, the number of students enrolled at a certain university has been decreasing.  Each year there is a 12% decrease in student enrollement.  Currently, 14,286 students are enrolled.  If this trend continues, how many students will be enrolled in 5 years?

Possible Answers:

Correct answer:

Explanation:

This is an exponential decay problem.  The formula for exponential decay is:

Where

= future value

= present value

= rate of decay

= number of periods

This problem requests the number of students five years in the future.  The rate of decay is twelve percent.  Therefore:

### Example Question #2 : Radioactive Decay Equations

Sceintists recently discovered a new type of metal compound. They have roughly 15 grams of this compound, which has a half life of 16 hours. Approximately how much of this substance will the scientists have in 24 hours?

Possible Answers:

grams

grams

grams

grams

grams

Correct answer:

grams

Explanation:

Recall the radioactive decay formula:

The half life formula is:

, where  is the half life.

Plug in the given half life:

Plug this value into the radioactive decay formula:

grams

### Example Question #3 : Radioactive Decay Equations

The equation for radioactive decay is,

.

Where  is the original amount of a radioactive substance,  is the final amount,  is the half life of the substance, and  is time.

The half life of Carbon-14 is about  years. If a fossile contains  grams of Carbon-14 at time , how much Carbon-14 remains at time  years?

Possible Answers:

None of the other answers.

Correct answer:

Explanation:

Using the equation for radioactive decay, we get:

### Example Question #4 : Radioactive Decay Equations

The number of fish in an aquarium is decreasing with exponential decay. The population of fish is decreasing by  each year. There are  fish in the aquarium today. If the decay continues how many fish will be in the aquarium in  years?

Possible Answers:

None of these answers are correct

Correct answer:

Explanation:

Every year the population of fish losses 7%. In other words, every year 93% of the fish remain from the previous year. Knowing this, we can use the original number of fish to find the number of fish for the next year. Since we want to know the number of fish 4 years from now, we multiply 1500 by 93% four times.

### Example Question #5 : Radioactive Decay Equations

The population of a city is decreasing. The city has a population of , people today, but the population decreases by  every year. What will be the population of the city in  years if this continues?

Possible Answers:

Correct answer:

Explanation:

Because the population of the city is decreasing every year at 10.5% we can find the population after each year by using

Because this decrease will continue every year for the 6 years, we can continue to multiply the population by the decay for every year.

### Example Question #6 : Radioactive Decay Equations

There is water leaking out of a cup.  of the water is leaking out every minute. How many kilograms of water will be left in  minutes and  seconds, if there are  kilograms of water ,, in the cup right now?

Possible Answers:

kilograms

kilograms

None of these answers are correct

kilograms

Correct answer:

kilograms

Explanation:

Because the water is leaking at a continuous rate, we can use the exponential decay equation.

is the decay of the problem, 12% or 0.12.  is equal to how many times the water will have a 12% decay. This can be calculated as

To calculate this we must first convert both time to seconds

Our equation is then

### Example Question #7 : Radioactive Decay Equations

An animal population is dying out. There is a decrease in this number of animals by  every year. In  years, there will be  of this animal left. What is the current population of this animal today?

Possible Answers:

Correct answer:

Explanation:

This is an exponential decay problem. Therefore, we can use this equation.

is the animal population after the 7 years.  is the animal population right now.  is the decay of the animal population every year.  is the time period of the animal populations decay.

From the problem we know after the 7 years the animal population will be 80, so

The population is decreasing by 8% every year, therefore

= 8% = 0.08

is equal to the number of times a decay of 8% has occurred. Since an 8% decay happens every year, and the population is 7 years from now, the population has decayed 8%, 7 times. In other words,

These values give us,

Rearranging this equation we can solve for

### Example Question #8 : Radioactive Decay Equations

A school is losing a certain number of students each year. This year, the school has  students. Four years ago the school had  students. The yearly rate of the school losing students has been the same for the last four years. What is the school's yearly rate of losing students?

Possible Answers:

Correct answer:

Explanation:

This is an exponential decay problem, meaning that the decay of the school's students can be found using

is the number of students currently at the school, which is 242

is the number of students that were at the school 4 years ago, which is 591

is the number of times the decay has occurred. Since, we are trying to find the yearly decay, the decay that happened to the school from one year to the next, and we have the number of students from 4 years ago,

is the decay rate of the school that we are trying to find. Because we have every number except for , we can plug the values into the equation to solve for .

= 20%

### Example Question #9 : Radioactive Decay Equations

Cells in a dish have started to decay. The cells are decaying by  every  minutes. When you left the cells there were  cells in the dish. Now there are  cells in the dish. Approximately how long did you leave the cells for?

Possible Answers:

Correct answer:

Explanation:

This is an exponential decay problem. That means that after 20 minutes 3% of the cells decay and 97% of the cells in the dish are left. To find the new amount of cells in the dish, we multiple the original number by the 97%.

The number of cells after every 20 minute interval can be calculated this way. Therefore, to find how long the cell were decaying we use,

Which can be rewritten as,

Now we can solve for , which is the amount of time that you were gone

To solve for , we must take the log of both sides to base 10. This will give us,

Remember!  is the number of decays that the cells went through. Each decay took 20 minutes to get through, however.

Therefore, the time that you were gone is

### Example Question #10 : Radioactive Decay Equations

Suppose 5 milligrams of element X decays to 3.2 milligrams after 48 hours.  What is the decay rate on a day to day basis?

Possible Answers:

Correct answer:

Explanation:

Write the formula for radioactive decay.

Substitute the values in the equation and solve for lambda.

Divide by 3.2 on both sides.

Take the natural log of both sides to eliminate the exponential.

Divide by negative two on both sides.

Convert this to a percentage.

This element's decay rate is approximately:

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