Exponential and Logarithms Real world models

Exponential and Logarithm functions KUS objectives BAT Solve real life problems involving growth functions of the form y = Aebx+c

Real life problems: growth functions Think Pair Share WB15 The Price of a used car is given by the formula: 𝑃=16000 𝑒 − 𝑡 10 Calculate the value of the car when it is new Calculate the value after 5 years c) What is the implied value of the car in the long run (ie – what value does it tend towards?) d) Sketch the Graph of P against t P = 16000e - t 10 a) Calculate the value of the car when it is new  The new price implies t, the time, is 0…  Substitute t = 0 into the formula… P = 16000e - 0 10 P = 16000e P = £16000

The Price of a used car is given by the formula: 𝑃=16000 𝑒 − 𝑡 10 WB15 solution The Price of a used car is given by the formula: 𝑃=16000 𝑒 − 𝑡 10 b) Calculate the value after 5 years…  5 years implies t = 5  Substitute t = 5 into the formula… P = 16000e - t 10 P = 16000e - 5 10 -0.5 P = 16000e P = £9704.49 3A

WB15 solution The Price of a used car is given by the formula: 𝑃=16000 𝑒 − 𝑡 10 c) What is the implied value of the car in the long run (ie – what value does it tend towards?)  Imagine t tends towards infinity (gets really big) P = 16000e - t 10 P = 16000 x 0 P = £0 1 (10√e)t e - t 10 Bigger t = Bigger denominator = Smaller Fraction value… 3A

The Price of a used car is given by the formula: 𝑃=16000 𝑒 − 𝑡 10 WB15 solution P The Price of a used car is given by the formula: 𝑃=16000 𝑒 − 𝑡 10 d) Sketch the Graph of P against t  Value starts at £16000  Tends towards 0, but doesn’t get there… £16000 P = 16000e - t 10 t  t is independent so goes on the x axis  P is dependant on t so goes on the y axis 3A

Calculate the number of elephants in the herd in 2003 Think Pair Share WB16 The number of elephants in a herd can be represented by the equation: Where n is the number of elephants and t is the time in years after 2003. Calculate the number of elephants in the herd in 2003 Calculate the number of elephants in the herd in 2007 Calculate the year when the population will first exceed 100 elephants What is the implied maximum number in the herd? Calculate the number of elephants in the herd in 2003  Implies t = 0 t = 0 e0 = 1

WB16 solution b) Calculate the number of elephants in the herd in 2007  Implies t = 4 t = 4 Round to the nearest whole number c) Calculate the year when the population will first exceed 100 elephants  Implies N = 100 Subtract 150 Divide by -80 Take natural logs 2003 + 19 = 2022 Multiply by 40

WB16 solution d) What is the implied maximum number in the herd?  Implies t  ∞ Rearrange As t increases Denomintor becomes bigger Fraction becomes smaller, towards 0

WB17 In 1847, a group of pioneering UK Farmers headed off to a beautiful Caribbean island to make a new life. After t years, the population P of the farmers was given by 𝑃=236 𝑒 0.01𝑡   a) State the initial population of the farmers. b) Sketch the graph of P against t. c) State the population of farmers after 25 years. d) Find out after how many years it takes for the population to exceed 5000. a) When 𝑡=0, 𝑃=236 𝑒 0 =236 c) When 𝑡=25, 𝑃=236 𝑒 0.25 =… d) When 𝑃=5000, 236 𝑒 0.01𝑡 =5000 ln 236 𝑒 0.01𝑡 = ln 5000 ln 236 + ln 𝑒 0.01𝑡 = ln 5000 0.01𝑡= ln 5000 − ln 236 𝑡=100 ln 5000 − ln 236 = …

One thing to improve is – KUS objectives BAT Solve real life problems involving growth functions of the form y = Aebx+c self-assess One thing learned is – One thing to improve is –

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