MATH IN SOCIETY Growth Models

Lesson Objective To gain an understanding of the 2 fundamental growth models Arithmetic (algebraic/linear) Geometric (exponential)

Linear (Algebraic) Growth Linear growth means that an initial value grows by the same amount in each time step. Ex. Your corn height measures 5 inches on Monday morning, 8 inches on Tuesday morning and then 11 inches on Wednesday morning and so on. So it is growing by 3 inches a day. This could be graphed with the Time on the x-axis and the Height on the y-axis. Called a recursive relationship b/c the next value in a sequence is related to the previous values.

Linear (Algebraic) Growth Formula If a quantity starts at size P­ 0 and grows by d, the common difference, every time period, then the quantity after n time periods can be determined using either of these relations: Recursively: P­ n = P­ n-1 + dsimple step model Explicitly:P­ n = P­ 0 + d*nlong prediction model Seen it before? d = slope P­ n = P­ 0 + d*n is the same as y = mx + b or (b + mx) Note: consider the validity of the data model – does it always hold?

Linear (Algebraic) Growth Ex. The population of elk in a national forest was measured to be 12,000 in 2003 (n=0), and was measured again to be 15,000 in 2007 (n=4). If the population continues to grow linearly at this rate, what will the elk population be, 11 years later, in 2014? Solve: P­ n = P­ 0 + d*n for P n Calculate d (slope) from the 2 ordered pairs Po = 12,000 in 2003 P 4 = 15,000 in years later. P­ 4 = P­ 0 + d*4 → = d d = 3000/4 = 750 per year P­ n = P­ 0 + d*n = *( ) P­ n = 20,250 elk 11 years are the initial population measurement Question to ponder: does this model hold forever?

Linear (Algebraic) Growth Ex. The cost, in dollars, of a gym membership for n months can be described by the explicit equation P­ n = n. What does this equation tell us? What would be the 6 month cost? Solve: $70 (P o ) is probably the up-front joining fee $30 (d) is the monthly (n) fee For 6 months (n) gym cost P n = $ *6 P n = $250

Exponential (Geometric) Growth Exponential growth refers to the situation where successive changes in a population differ by a constant ratio, like % (as distinct from a constant amount for arithmetic change). Examples: Compound interest at a constant interest rate provides exponential growth of the capital. A virus (for example SARS, or smallpox) typically will spread exponentially at first, if no artificial immunization is available. Each infected person can infect multiple new people.

Exponential (Geometric) Growth Formula. If a quantity starts at size P­ 0 and grows by R% (written as a decimal, r) every time period, then the quantity after n time periods can be determined using either of these relations: Recursively:P­ n = (1+r)*P­ n-1 Explicitly:P­ n = P ­0 *(1+r) n r is the growth rate, % as decimal (1 + r) is the growth multiplier

Exponential (Geometric) Growth Ex. India is the second most populous country in the world, with a population in 2008 of about 1.14 billion people (P o ). The population is growing by about 1.34% (r) each year. If this trend continues, to what value (P n ) will India’s population grow by 2020? Solve: P ­n = P ­ 0 (1+r) n (what is n in this case?) n = 0 in 2008, so n = (2020 – 2008) = 12 in 2020 P ­n = 1.14( ) 12 P ­n = million

Exponential (Geometric) Growth *

Exponential (Geometric) Growth Follow-on: In 1990 (n = 0), residential energy use in the US was responsible for 962 million (P o ) metric tons of carbon dioxide emissions. By the year 2000 (n = 10), that number had risen to 1182 million (P n ) metric tons. If the emissions grow exponentially at 2.08%, what will be the emissions tonnage by 2050, 60 years after the initial recording? Solve: P ­n = P­ 0 (1+r) n P ­n = 962( ) 60 P ­n = 3,308.4 million metric tons!

Exponential (Geometric) Growth How big is 3,308.4 million metric tons? 3,646 million US tons In terms of school buses, that weigh about 15 tons? ~245 million school buses In terms of the Empire State Building, that weighs about 365,000 tons? ~10,000 Empire State Bldgs

Exponential (Geometric) Growth Looking back at the last example, for the sake of comparison, what would the carbon emissions be in 2050 if emissions grow linearly at the same rate? Solve: P­ 10 = P ­0 + 10d linear growth formula, find d 1182 = d 220 = 10d d = 22 million metric tons per year In 2050 (60 years from the initial recording) P 60 = (60) = 2,282 million metric tons ~6,000 Empire St Bld

Recognizing Arithmetic vs Geometric growth Arithmetic will indicate either explicitly or implicitly a fixed amount per time period Ex. Every year the population increases by 2000 people Geometric will indicate either explicitly or implicitly a percentage (%) growth rate Ex. Every year the population increases by 3%

Formula Summary – Linear & Exponential Linear growth P­ n = P­ 0 + d*nP­ o = initial value, d = amt of change, n = time Exponential growth P ­n = P­ 0 (1+r) n r = rate as decimal

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