More Applications of Exponential Functions

Applications of exponential functions occur in the physical, life, and social sciences, as well as in the world of business.

A culture begins with 100 cells. Observation shows that each cell divides into 2 cells every 45 minutes. How many cells will be in the culture after 24 hours?

The initial amount, 100, will be the leading coefficient, because this is the number of cells present at time 0 (and 20 = 1). The next thing to determine is the number of cell divisions that take place. The cells divide every 45 minutes, so you need to determine how many 45-minute periods exist in 24 hours; you find that there are (24)(60)/45 = 32. The number of cells present at the end of 24 hours will be 100(2)32 = 429,296,729,600. (Using the analysis written, one can find the number of cells after m minutes by the formula

.)

Data supplied by the Population Reference Bureau showed the United States population to be 307 million people in 2009 with a growth rate of 0.876% over the next 41 years. The equation modeling the U.S. population is P(t) = 307(1.00876)t, where t represents the number of years since 2009. On the basis of this model, predict the population of the United States in 2020.

2020 is 11 years after 2009, so the U.S. population for 2020 is predicted to be P(11) = 307(1.00876)11 = 337.913 million people.

Iodine-131 is a radioactive element used in medical treatments and has a half-life of 8.0197 days. If an initial dose of 10 milligrams (mg) is injected into a body, the amount of iodine-131 remaining after d days is given by the equation

. How many milligrams of I-131 will remain in the body after 12 days?

Government regulations require banks to have a specified amount of cash available at all times. When a bank falls short of this amount, it borrows a sum of money from another institution for a short period of time. In such cases, the interest is computed continuously. Suppose Bank A needs to borrow $2,500,000 from Bank B for 3 hours at a rate of 1.9% compounded continuously. The formula for the amount that the borrower will owe to the lender in such a transaction is A = 2500000e0.019t, where t is the amount of time in years. How much must Bank A pay in interest?

3 hours represents 1/8 of a day, or (1/8)(1/365) of a year. The amount of money Bank A must pay Bank B is A = 2500000e(0.019)(1/8)(1/365) = 2,500,016.27. Bank A will owe Bank B $16.27 in interest.

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