5. 6. Further Applications and Modeling with 5.6 Further Applications and Modeling with Exponential and Logarithmic Functions Physical Science Applications: A0 is some initial quantity t represents time k > 0 represents the growth constant, and k < 0 represents the decay constant

5.6 Age of a Fossil using Carbon-14 Dating Example Carbon 14 is a radioactive form of carbon found in all living plants and animals. After a plant or animal dies, the radiocarbon disintegrates. Scientists determine the age of the remains by comparing the amount of carbon 14 present with the amount found in living plants and animals. The amount of carbon 14 present after t years is given by Find the half-life. Solution Let Divide by A0.

5.6 Age of a Fossil using Carbon-14 Dating The half-life is 5700 years. Take the ln of both sides. ln ex = x and quotient rule for logarithms Isolate t. Distribute and use the fact that ln1 = 0.

5.6 Finding Half-life Example Radium-226, which decays according to has a half-life of about 1612 years. Find k. How long does it take a 10-gram sample to decay to 6 grams? Solution The half-life tells us that A(1612) = (½)A0.

5.6 Finding Half-life Thus, radium-226 decays according to the equation Now let A(t) = 6 and A0 = 10 to find t.

5.6 Financial Applications Example How long will it take $1000 invested at 6% interest compounded quarterly to grow to $2700? Solution Find t when A = 2700, P = 1000, r = 0.06, and n = 4.

5.6 Amortization Payments A loan of P dollars at interest rate i per period may be amortized in n equal periodic payments of R dollars made at the end of each period, where The total interest I that will be paid during the term of the loan is

5.6 Using Amortization to Finance an Automobile Example You purchase a camper trailer for $24,000. After a down payment of $4000, the balance will be paid off in 36 equal monthly payments at 8.5% interest per year. Find the amount of each payment. How much interest will you pay over the life of the loan? Solution

5.6 Financial Applications Example World population in billions during year x can be modeled by an exponential function. Solve the equation below to estimate the year when world population reached 7 billion. Solution