Using calculus, it can be shown that when the rate of growth or decay of some quantity at a given instant is proportional to the amount present at that instant; it is described by… Q(t) = The quantity at any given time Q 0 = The initial amount k = the constant of proportionality t = time

We say Q grows exponentially if k > 0 and decays exponentially if k < 0. The constant k is called the constant of proportionality Ex 1: A culture of cells is observed to triple in size in 2 days. Assuming exponential growth, how large will the culture be in 5 days? Solution: Exponential growth/decay problems are two step problems. Step 1: Find k Start with the time that it takes for the quantity to double, triple etc. t = 2 We know that after 2 days the cells triple in size. Therefore, Solve for k by taking the natural log of both sides.

2k = ln 3 Step 2: Substitute the value of k back into the equation and solve for what the question asks. Evaluate e and its exponent Q(5) = Q 0 After five days the culture of cells will be times larger than it originally was.

Ex 2: The following table gives the population in the United States for the years 1930 through Assume the population changes by an amount that is proportional to the amount of population present. Predict the population in the years 1990 and 2000 using the following population figures. a.) 1930 and 1940b.) 1970 and 1980 Solution:a.) Again, we must find the value of k first. We have a beginning time (1930) and an ending time (1940) YearPopulation (millions)

Now, use the formula you just created to find the population in 1990 and : t = 60 because it is 60 years from 1930 to : b.)We do the exact same procedure as in part a except we use the years 1970 and :2000:

If you were to graph the information from the table, part a and part b on the same coordinate plane you would see that part b has a graph that is closest to the actual information. Reason: use the most current data that you have to solve problems. It will better represent the actual amounts. Ex 3: The radioactive isotope strontium 90 has a half-life of 29.1 years, which is the time that it takes for one half of the original amount to decay to another substance. a.) How much strontium 90 will remain after 20 years from an initial amount of 300 killograms? b.) How long will it take for 80% of the original amount to decay. Solution:Find k.We know that after 29.1 years that only have of the Original amount will still remain.

Notice that k < 0. Recall that if k< 0 we are decaying exponentially. Now, substitute the value of k into the formula. Now, we are ready to answer questions a and b. a.) t = 20; b.)The question wants to know when 80% will be gone. We could also re-word this by saying: When will 20% be left? We must do it this way since the formula always represents the amount of substance left. Find 20% of the original amount (300 kg) 0.20(300) = 60 kg = Q(t)

Ex 4: Estimate the age of the Ice Man assuming that 54.6% of the original amount of Carbon 14 remained at the time of the discovery. Carbon 14 has a half-life of 5730 years. Solution:

Since we are looking for time we must have a value for Q(t). The iceman died about 5000 years ago.

Newton’s Law of Cooling This law predicts the temperature T(t) of an object. Ex 5: A well-cooked turkey with uniform temperature 170° is taken out of an oven and placed in a 70° room. After 5 minutes, the turkey’s temperature has decreased to 160°. What temperature would we expect the bird to be after another 15 minutes? Solution: Find k first!