Write a function that gives the population of the colony at any time , in days. Hint: Express the values you calculated in part (1) using powers of . What connection do you see between the value of and the exponent on ?
Graph your function from part (3) using technology. (Use the table to choose an appropriate window.) The graph should resemble your hand-drawn graph from part (2).
The function in Investigation 7.1.1 describes exponential growth. During each time interval of a fixed length, the population is multiplied by a certain constant amount. In this case, the bacteria population grows by a factor of 3 every day.
Researchers often use cell lines from the fruit fly Drosophila melanogaster to study protein interactions related to cancer and other diseases. From 60% to 70% of human disease genes are found in Drosophila cells, and gene discoveries in the flies have led to parallel studies in vertebrates.
One milliliter of culture contains about 1 million Drosophila cells, and the population doubles every 24 hours. The table shows the population, , of Drosophila cells, in millions, as a function of time in days.
Looking at the table, we see that we multiply the fruit fly population by 2 every day, so that after days, the initial population is multiplied by . Because the population grows by a factor of 2 each day, the function describes exponential growth. We can express functions that describe exponential growth in a standard form.
For the Drosophila cell population, the growth factor is , and the initial value is million cells, so we have
There is a sort of similarity between the formula for exponential functions and the formula for linear functions. Each has an initial value and a constant that describes change. But compare the graph of exponential growth, described by a constant growth factor, with linear growth. You can see that the graph of the fruit fly population is not a straight line with a constant slope, as a linear function would be.
Once again, in the examples above, you can see that the graphs of these exponential functions are not linear. In each case, the function grows slowly at first, but eventually grows faster and faster.
is a linear function with -intercept 5 and slope 2; is an exponential function with initial value 5 and growth factor 2. In a way, the growth factor of an exponential function is analogous to the slope of a linear function: Each measures how quickly the function is increasing.
However, for each unit increase in , 2 units are added to the value of , whereas the value of is multiplied by 2. An exponential function with growth factor 2 eventually grows much more rapidly than a linear function with slope 2, as you can see by comparing the graphs or the function values in the tables.
A solar energy company sold $80,000 worth of solar collectors last year, its first year of operation. This year its sales rose to $88,000. The marketing department must estimate its projected sales for the next 3 years.
If the marketing department predicts that sales will grow linearly, what sales total should it expect next year? Graph the projected sales figures over the next 3 years, assuming that sales will grow linearly.
If the marketing department predicts that sales will grow exponentially, what sales total should it expect next year? Graph the projected sales figures over the next 3 years, assuming that sales will grow exponentially.
Let represent the company’s total sales years after starting business, where is the first year of operation. If sales grow linearly, then has the form . Because , the intercept is 80,000. The slope of the graph is
dollars year dollars/year
where is the increase in sales during the first year. Thus, , and sales grow by adding $8000 each year. The expected sales total for the next year is
Let represent the company’s sales assuming that sales will grow exponentially. Then has the form , and the initial value is . We find the growth factor in sales over the first year by dividing by :
so
Thus, , and the expected sales total for the next year is
In the examples above, exponential growth was modeled by increasing functions of the form
where the growth factor, , is a number greater than 1. If we multiply the function value by a number smaller than 1, t he function values will decrease. [TK] Thus, if , then is a decreasing function. In this case, we say that the function describes exponential decay, and the constant is called the decay factor.
A small coal-mining town has been losing population since 1940, when 5000 people lived there. At each census thereafter (taken at 10-year intervals), the population declined to approximately 0.90 of its earlier figure.
Graph your function from part (3) using technology. (Use the table to choose an appropriate window.) The graph should resemble your hand-drawn graph from part (2).
Before the introduction of disposable containers, soft drinks and draught beer were sold in refillable glass botles. During the second half of the last century, the percent of beer volume sold in refillable glass bottles declined to 0.942 of its previous value each year.
In 1944, 98% of beer was sold in refillable bottles. Write a formula for the percent of beer sold in refillable bottles as a function of , the number of years after 1944.
Exponential growth is often described as growth by a certain percent increase. Suppose the town of Lakeview had 4000 residents in the year 2000, and grew at a rate of 5% per year. This means that each year we add 5% of last year’s population to find the current population, . Thus
In 2000, In 2001, Add 5 of In 2002, Add 5 of
and so on. Now here is the important observation about percent increase:
A formula for the population of Lakeview years after 2000 is
This formula describes exponential growth with a growth factor of . In general, a function that grows at a percent rate , where is expressed as a decimal, has a growth factor of .
The percent increase in the cost of bread is 12% every year. Therefore, the growth factor for the cost of bread is every year. If represents the price of bread after years, then , and we multiply the price by 1.12 every year, as shown in the table.
Tombstone, Arizona was the most famous "boomtown" during the gold rush in the American west. It was established in December, 1879, after the discovery of a large silver deposit nearby. The original town had 40 dwellings and a population of 100. Over the next two to three years, the population grew at an average rate of 19% per month.
What was the population one year later, in December, 1880?
Compound interest is another example of exponential growth. Suppose you deposit a sum of money, , into an account that pays 5% interest compounded annually. "Compounded" means that each year your interest, 5% of your current balance, is added to your account, so your balance, , grows by a factor of 1.05.
As an example, suppose you put $100 in an account that pays 5% interest compounded annually. Using the formula above, we can make a table showing the balance in your account over the next few years.
We have seen that a percent increase of (in decimal form) corresponds to a growth factor of . A percent decrease of corresponds to a decay factor of . For example, if a population declines by 25% each year, then each year the new population is 75% of its previous value. So
and . Remember that multiplying by gives us the population remaining, not the amount of decline. [TK]
According to Context magazine: "Computing prices have been falling exponentially for the past 30 years and will probably stay on that curve for another couple of decades." In fact, prices have been falling at a rate of 37% every year. Suppose an accounting firm invests $50,000 this year in new computer equipment.
In the preceding Example, the cost of computer equipment decreases by 37% each year, so 63% of the value remains, and the decay factor for the value function is 0.63, not 0.37. The function gives the cost remaining, not the amount that has declined.
A new car begins to depreciate in value as soon as you drive it off the lot. Some models depreciate linearly, and others depreciate exponentially. Suppose you buy a new car for $20,000, and 1 year later its value has decreased to $17,000.
The value of your stock portfolio fell 10% last year, but this year it increased by 10%. How does the current value of your portfolio compare to what it was two years ago?
Sales of Windsurfers have increased 12% per year since 2010. If Sunsails sold 1500 Windsurfers in 2010, how many did it sell in 2015? How many should it expect to sell in 2022?
Paul bought a house for $200,000 in 1983. For the next 20 years, housing prices rose an average of 5% per year. How much was the house worth in 1995? In 2000?
A typical beehive contains 20,000 insects. The population can increase in size by a factor of 2.5 every 6 weeks. How many bees could there be after 4 weeks? After 20 weeks?
During a vigorous spraying program, the mosquito population was reduced to of its previous size every week. If the mosquito population was originally estimated at 250,000, how many mosquitoes remained after 3 weeks of spraying? After 9 weeks?
Scuba divers find that the water in Emerald Lake filters out 15% of the sunlight for each 4 feet they descend. How much sunlight penetrates to a depth of 20 feet? To a depth of 45 feet?
Plutonium-238 is a radioactive element that decays over time into a less harmful element at a rate of 0.8% per year. A power plant has 50 pounds of plutonium-238 to dispose of. How much plutonium-238 will be left after 10 years? After 100 years?
In the 1940s David Lack undertook a study of the European robin. He tagged 130 one-year-old robins and found that on average 35.6% of the birds survived each year. (Source: Burton, 1998.)
According to the data, how many robins would have originally hatched to produce 130 one-year-olds?
Many insects grow by discrete amounts each time they shed their exoskeletons. Dyar’s rule says that the size of the insect increases by a constant ratio at each stage. (Source: Burton, 1998.)
Dyar measured the width of the head of a caterpillar of a swallowtail butterfly at each stage. The caterpillar’s head was initially approximately 42 millimeters wide and was 63.84 millimeters wide after its first stage. Find the growth ratio.
A researcher starts 2 populations of fruit flies of different species, each with 30 flies. Species A increases by 30% in 6 days, and species B increases by 20% increases in 4 days.
What was the population of species A after 6 days? Find the daily growth factor for species A.