This doubling time is illustrated in the following applet. Doubling time and half life. The blue crosses and lines highlight points at which the population size has double or shrunk in half; you can move these points by dragging the blue points. You can drag the blue crosses to change the intervals. You can click the arrows to change the scales of the graph. The exponential growth function can be written in the form.
A field has an initial population of 8 bunnies. Bunnies reproduce very quickly, and a reasonable estimate is that the bunny population doubles five times every year. Assuming there are no predators to reduce the population, how many bunnies will be in the field after four years? We can use the doubling form of the exponential function where our initial value is 8 and our doubling time is 0.
After four years, the population will have grown from 8 bunnies to nearly 8. If we know how long it takes for the value to double, we can describe the exact same growth model as. Exponential decay is the same as exponential growth except we repeatedly multiply by a factor that is between 0 and 1, so the result shrinks over time.
If we know how long it takes for the value to be cut in half, we can describe the exact same growth model as. We measured exponential growth using the "doubling-time", and we can measure exponential decay using the "halving-time. In fact, the term half-life is often preferred over the more awkward "halving-time" even in situations that have nothing to do with lifetimes.
We will investigate exponential decay in terms of temperature. When an object is at the same temperature as its environment, no heat flows at all.
When objects cool down, their temperature decays exponentially. Leaving food out too long at room temperature can cause harmful bacteria to grow to dangerous levels that can cause illness. This range of temperatures is often called the "Danger Zone. Simply placing a container of hot soup into a refrigerator will cool the food too slowly, so restaurants typically use ice-water baths before refrigeration. If the "half-life" of the exponential decay is 3. The cooling period of six hours is almost long enough for two "half-lifes" of 3.
So the soup might be safe. To know for sure, we use the exponential decay formula. No, the soup will not cool in less than six hours. The proposed cooling method is not safe because the exponential decay is too slow. One solution would be to split the soup into several smaller containers that would cool more swiftly. There is a handy approximation that relates the percent change to the characteristic time.
How old is a skull that contains one-fifth as much radiocarbon as a modern skull? Note that the half-life of radiocarbon is years. The population of Cairo grew from 5 million to 10 million in 20 years. Use an exponential model to find when the population was 8 million. Starting from 8 million New York and 6 million Los Angeles , when are the populations equal?
Starting from when will. The effect of advertising decays exponentially. If at and at what was at. If at and at when does. If a bank offers annual interest of 7. What continuous interest rate has the same yield as an annual rate of. You are trying to save in 20 years for college tuition for your child. If interest is a continuous how much do you need to invest initially? You are cooling a turkey that was taken out of the oven with an internal temperature of After 10 minutes of resting the turkey in a apartment, the temperature has reached What is the temperature of the turkey 20 minutes after taking it out of the oven?
You are trying to thaw some vegetables that are at a temperature of To thaw vegetables safely, you must put them in the refrigerator, which has an ambient temperature of You check on your vegetables 2 hours after putting them in the refrigerator to find that they are now Plot the resulting temperature curve and use it to determine when the vegetables reach.
You are an archaeologist and are given a bone that is claimed to be from a Tyrannosaurus Rex. You know these dinosaurs lived during the Cretaceous Era million years to 65 million years ago , and you find by radiocarbon dating that there is 0. Is this bone from the Cretaceous? The spent fuel of a nuclear reactor contains plutonium, which has a half-life of 24, years.
If 1 barrel containing 10kg of plutonium is sealed, how many years must pass until only of plutonium is left? For the next set of exercises, use the following table, which features the world population by decade. Where is it increasing and what is the meaning of this increase? The population is always increasing. Using your previous answers about the first and second derivatives, explain why exponential growth is unsuccessful in predicting the future.
For the next set of exercises, use the following table, which shows the population of San Francisco during the 19th century. Where is it increasing? What is the meaning of this increase? Is there a value where the increase is maximal? Skip to content 6. Applications of Integration.
Learning Objectives Use the exponential growth model in applications, including population growth and compound interest. Explain the concept of doubling time. Explain the concept of half-life. Exponential Growth Model Many systems exhibit exponential growth. Rule: Exponential Growth Model Systems that exhibit exponential growth increase according to the mathematical model.
Figure 1. An example of exponential growth for bacteria. Population Growth Consider the population of bacteria described earlier. Solution We have Then. Solution There are 81,, bacteria in the population after 4 hours. Hint Use the process from the previous example. Solution We have. If the initial size of the tumor is four cells, how many cells will there in three years?
In seven years? To calculate the number of cells in the tumor, we use the doubling time model. Change the time units to be the same. What is the approximate annual growth rate of the city? By solving the doubling time model for the growth rate, we can solve this problem. The half-life of a material is the time it takes for a quantity of material to be cut in half. This term is commonly used when describing radioactive metals like uranium or plutonium. For example, the half-life of carbon is years.
If a substance has a half-life, this means that half of the substance will be gone in a unit of time. If there are 40 grams present now, how much is left after three days? We want to find a model for the quantity of the substance that remains after t days. The amount of time it takes the quantity to be reduced by half is eight days, so this is our time unit.
After t days have passed, then t8 is the number of time units that have passed. Starting with the initial amount of 40, our half-life model becomes:. Lead is a radioactive isotope. It has a half-life of 3.
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