A 6kg quantity of radioactive isotope decays to 4 kg after 16 years. Find the decay constant of the isotope. k: y; The atmospheric pressure P(h) (in pounds per square inch) at a height h (in miles) above sea level on Earth satises a differential equation P' = kP for some positive constant k. (a) Measurements with a barometer show that P(0) = 1.7 and P(4) = 0.2. What is the decay constant k? (b) Determine the atmospheric pressure 14 miles above sea level. (a) k = i. (b) P(14) = I: A bank pays interest at a rate of 9%. What is the yearly multiplier if interest is compounded (a) yearly? (1)) three times a year? (c) continuously? Assume the world population will continue to grow exponentially with a growth constant k = 0.0132 (corresponding to a doubling time of about 52 years), it takes % acre of land to supply food for one person, and there are 13,500,000 square miles of arable land in in the world. How long will it be before the world reaches the maximum population? Note: There were 6.06 billion people in the year 2000 and 1 square mile is 640 acres. Answer: The maximum population will be reached some time in the year 9" Hint: Convert .5 acres of land per person (for food) to the number of square miles needed per person. Use this and the number of arable square miles to get the maximum number of people which could exist on Earth. Proceed as you have in previous problems involving exponential growth. Exponential Growth and Decay: Here are a couple of examples for applications of exponentials and logarithms. You nd out that in the year 1800 an ancestor of yours invested 100 dollars at 6 percent annual interest, compounded yearly. You happen to be her sole known descendant and in the year 2005 you collect the accumulated tidy sum of i, dollars. You retire and devote the next 10 years of your life to writing a detailed biography of your remarkable ancestor. Strontium90 is a biologically important radioactive isotope that is created in nuclear explosions. It has a half life of 28 years. To reduce the amount created in a particular explosion by a factor 1,000 you would have to wait y; years. Round your answer to the nearest integer. Seeds found in a grave in Egypt proved to have only 84% of the Carbon14 of living tissue. Those seeds were harvested y; years ago. The halflife of Carbon14 is 5,730 years