Odd one out? WHY?

Aims: To know the general functions for exponential growth and decay. To be able to solve questions about exponential growth and decay. Growth and Decay Lesson 1

t 0 f(t)f(t) Exponential growth In general, exponential growth can be modelled by the function f ( t ) = Ae kt A is the original quantity (the quantity when t = 0), f ( t ) is the quantity after time t and where t is time, k is a positive constant (the growth rate) y = Ae kt An exponential growth curve has the following basic shape: A Time can’t really be n_____________! Exponential g________ occurs when a quantity increases at a rate that is proportional to its size. For example, population growth.

Exponential growth Example The number of Rats N is increasing exponentially. Given that at time t = 0, N = 30 million and at time t = 20, N = 40 million, find: (a)The number of rats, N, when t = 30 (b)The value of t when there are 90 million rats Start with the general formula: N = Ae kt When t = 0, N = 30 When t = 20, N = 40 (a)When t = 30 N = (b)When N = 90, 90 =

On w/b Example The growth of a population is modelled by P=5000e 0.04t. (a)What was the size of the population when it was first counted? (b)What is the population at time t=20? (c)Find the rate of increase at time t = 20, to the nearest whole number.

Exponential decay Exponential d_______ occurs when a quantity decreases at a rate that is proportional to its size. In other words, when a quantity decays exponentially, the smaller it becomes, the more slowly it decays. E.g. the value of a car as it depreciates, the rate at which an object cools (when the external temperature is constant). In general, exponential decay can be modelled by the function f ( t ) = Ae – kt t 0 y = Ae – kt An exponential decay curve has the following basic shape. Using the example of a cars value, what can you say happens as time t, tends to ∞ ? _____________________________ A f(t)f(t)

Exponential decay Forensic scientists can predict a recently murdered victim’s time of death from the temperature of the body. They do this by applying Newton’s law of cooling: A body is discovered at 8.30 pm. The body’s temperature is recorded as 30°C and room temperature as 20°C. One hour later, the temperature of the body is 29°C. Assuming that the room temperature remains constant throughout, estimate the time of death. d = ae – kt where d is the temperature difference (between the body and its surroundings), a and k are constants and t is the time that has passed since the body started to cool. For example, Have a go! 4 mins

Exponential decay Let t = 0 at 8.30 pm. And d = 30 – 20 = And using d = ae – kt : When t = 1, one hour later: d = 29 – 20 = We are using a –k as this is a decay problem. k = to 3SF Take natural logs on both sides: 10 = ae – k (0) = therefore a =

Exponential decay Substitute these values of k and a into d = ae – kt : d = If we assume that the murder victim had a normal body temperature of 36.9 °C when she died and that the room temperature was 20°C, then at the time of death: So = 10 e –0.105 t Solving this equation for t will give the time since the victim died. t = 0 at 8.30 pm, so t = t =  The victim died at about pm.

On w/b – Exam Qu Do exercise 6D page 76 qu 2 - 5