Greenwood press
Download 1.81 Mb. Pdf ko'rish
|
book-20600
EXPONENTIAL GROWTH
31 The growth of the rabbit population in Australia from 1859 –1864 modeled by an exponential growth equation. A graph depicting growth of the rabbit population in Australia from 1859 –1864. six-month year periods (x) rabbit population (y) 1859 0 24 1 24 • 2 = 48 1860 2 24 • 2 • 2 = 96 3 24 • 2 • 2 • 2 = 192 1861 4 24 • 2 4 = 384 5 24 • 2 5 = 768 1862 6 24 • 2 6 = 1,536 7 24 • 2 7 = 3,072 1863 8 24 • 2 8 = 6,144 9 24 • 2 9 = 12,288 1864 10 24 • 2 10 = 24,576 pal held for the customer. The return addition of interest payments to the princi- pal so that the interest amount can earn interest in later years is called compound interest. The growth factor in compound-interest problems is 1 plus the annual yield. So an investor who buys a $5,000 CD advertised at 6.5 percent annual yield will receive 5000(1 + .065) x after x years. After three years, this CD would be valued at 5000(1.065) 3 = $6,039.75. Banks may choose to compound interest more frequently. The banking version of the exponential growth formula is A = P (1 + r/n) nt , where A is the amount at the end of t years, P is the start- ing principal, r is the stated interest rate, and n is the number of periods per year that interest will be compounded. A typical CD will have interest compounded each quarter. Financial institutions can offer more-frequent compounding, such as monthly or daily. Some even offer continuous compounding, which has the formula A = P e rt , where A is the value of the investment at time t, P is the ini- tial principal, r is the interest rate, and e ≈ 2.7183. For a given interest rate, more frequent compounding yields a higher return, but that return does not increase dramatically as the compounding period moves from months to days to continu- ous. Because the number of compounding periods can affect the rate of return on an investment, federal law requires financial institutions to state the annual yield as well as an interest rate so that consumers can make easier comparisons among investment opportunities. Benjamin Franklin was one of the pioneers in the use of exponential growth models for money and population. In 1790, Franklin established a trust of $8,000. He specified that his investment should be compounded annually for 200 years, at which time the funds should be split evenly between the cities of Philadelphia and Boston, and used for loans to “young apprentices like himself.” Franklin anticipated that the fund would be worth $20.3 million after 200 years if the annual yield averaged 4 percent. However, the annual yield averaged about 3.4 percent, so $6.5 million was in the fund when it was dispersed to the two cities in 1990. Franklin established the practice of studying the American population by using exponential growth. He recognized that the warning of the Englishman Thomas Malthus (1766 –1834) that population under exponential growth would outstrip food sources might apply to the new country of the United States. Frank- lin urged that the growth of states and the entire country be tracked each year. Some historians contend that President Lincoln used exponential growth models 70 years after Franklin’s recommendation. Lincoln used censuses from 1790 to 1860 to predict that the population of the United States would be over 250 mil- lion in 1930. The population did not reach this figure until 1990. This shows that exponential functions can describe situations only as long as the growth factor remains constant. There are many factors such as economics, war, and disease that can affect the rate of population growth. When the Center for Disease Control identifies a new epidemic of flu, expo- nential growth functions describe the numbers of early cases of infection quite well. A good definition of epidemic is a situation in which cases of disease in- crease exponentially. However, as people build up immunization, the disease Download 1.81 Mb. Do'stlaringiz bilan baham: |
Ma'lumotlar bazasi mualliflik huquqi bilan himoyalangan ©fayllar.org 2024
ma'muriyatiga murojaat qiling
ma'muriyatiga murojaat qiling