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Applications: Decay OBJECTIVES  Find a function that satisfies dP/dt = – kP.  Convert between decay rate and half-life.  Solve applied problems involving exponential decay. 3.4

Slide Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley The equation shows P to be decreasing as a function of time, and the solution Shows it to be decreasing exponentially. This is exponential decay. The amount present at time t is P Applications: Decay

Slide Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley 3.4 Applications: Decay

Slide Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley THEOREM 10 The decay rate k and the half–life T are related by 3.4 Applications: Decay

Slide Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Example 2: Plutonium-239, a common product of a functioning nuclear reactor, can be deadly to people exposed to it. Its decay rate is about % per year. What is its half-life? 3.4 Applications: Decay

Slide Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Example 3: The radioactive element carbon-14 has a half-life of 5750 yr. The percentage of carbon-14 present in the remains of plants and animals can be used to determine age. Archaeologists found that the linen wrapping from one of the Dead Sea Scrolls had lost 22.3% of its carbon-14. How old was the linen wrapping? 3.4 Applications: Decay

Slide Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Example 3 (continued): 1 st find the decay rate, k. Then substitute the information from the problem and k into the equation 3.4 Applications: Decay

Slide Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Example 3 (concluded): 3.4 Applications: Decay

Slide Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Example 4: Following the birth of their granddaughter, two grandparents want to make an initial investment of P 0 that will grow to $10,000 by the child’s 20 th birthday. Interest is compounded continuously at 6%. What should the initial investment be? We will use the equation 3.4 Applications: Decay

Slide Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Example 4 (continued): Thus, the grandparents must deposit $ , which will grow to $10,000 by the child’s 20 th birthday. 3.4 Applications: Decay

Slide Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley THEOREM 11 The present value P 0 of an amount P due t years later, at an interest rate k, compounded continuously, is given by 3.4 Applications: Decay

Slide Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Newton’s Law of Cooling The temperature T of a cooling object drops at a rate that is proportional to the difference T – C, where C is the constant temperature of the surrounding medium. Thus, The function that satisfies the above equation is 3.4 Applications: Decay

Slide Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Example 6: Found stabbed by a collection of number 2 pencils, Prof. Cal Kulice’s body was slumped over a stack of exams with plenty of red marks on them. A coroner arrives at noon, immediately takes the temperature of the body, and finds it to be 94.6°. She waits 1 hr, takes the temperature again, and finds it to be 93.4°. She also notes that the temperature of the room is 70°. When was the murder committed? 3.4 Applications: Decay

Slide Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Example 6 (continued): We will assume that the temperature of the body was normal (T = 98.6°) when the murder was committed (t = 0). Thus, This gives 3.4 Applications: Decay

Slide Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Example 6 (continued): To find the number of hours, N, since the murder was committed, we must first find k. From the two temperature readings the coroner made, we have Then, we can solve for k. 3.4 Applications: Decay

Slide Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Example 6 (continued): 3.4 Applications: Decay

Slide Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Example 6 (concluded): Then we can substitute back into either one of our first equations to solve for N. 3.4 Applications: Decay