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

3.4 Applications: Decay 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 = 0 is P0.

3.4 Applications: Decay

3.4 Applications: Decay THEOREM 10 2012 Pearson Education, Inc. All rights reserved 3.4 Applications: Decay THEOREM 10 The decay rate k and the half–life T are related by

3.4 Applications: Decay Example 1: Plutonium-239, a common product of a functioning nuclear reactor, can be deadly to people exposed to it. Its decay rate is about 0.0028% per year. What is its half-life?

3.4 Applications: Decay Quick Check 1 a.) The decay rate of cesium-137 is 2.3% per year. What is its half-life? b.) The half-life of barium-140 is 13 days. What is its decay rate?

3.4 Applications: Decay Example 2: The radioactive element carbon-14 has a half-life of 5730 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 Example 2 (continued): 2012 Pearson Education, Inc. All rights reserved 3.4 Applications: Decay Example 2 (continued): 1st find the decay rate, k. Then substitute the information from the problem and k into the equation

3.4 Applications: Decay Example 2 (concluded):

3.4 Applications: Decay Quick Check 2 How old is a skeleton found at an archaeological site if tests show that it has lost 60% of its carbon-14? First find the decay rate. We know from Example 2 that the decay rate Then substitute the information from the problem and into

3.4 Applications: Decay Quick Check 2 Concluded Thus the skeleton is approximately 7575 years old.

3.4 Applications: Decay Example 3: Following the birth of their granddaughter, two grandparents want to make an initial investment of P0 that will grow to $10,000 by the child’s 20th birthday. Interest is compounded continuously at 6%. What should the initial investment be? We will use the equation

3.4 Applications: Decay Example 3 (continued): Thus, the grandparents must deposit $3011.94, which will grow to $10,000 by the child’s 20th birthday.

3.4 Applications: Decay Quick Check 3 Repeat Example 3 for an interest rate of 4% We will use the equation Thus the grandparents must deposit $4493.29, which will grow to $10,000 by the child’s 20th birthday.

3.4 Applications: Decay THEOREM 11 2012 Pearson Education, Inc. All rights reserved 3.4 Applications: Decay THEOREM 11 The present value P0 of an amount P due t years later, at an interest rate k, compounded continuously, is given by Note:

Newton’s Law of Cooling 3.4 Applications: Decay 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 Example 4: A body is found slumped over a desk in a study. 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 Example 4 (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 Example 4 (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 Example 4 (continued):

3.4 Applications: Decay Example 4 (concluded): Then we can substitute back into either one of our first equations to solve for N.

3.4 Applications: Decay Section Summary The decay rate, k, and the half–life, T, are related by The present value of an amount p due t years later, at an interest rate k, compounded continuously, is given by P0 = Pe–kt.

3.4 Applications: Decay Section Summary According to 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 surrounding room temperature. Thus, we have and