Chapter 4 Introduction to Mathematical Modeling Tpes of Modeling 1) Linear Modeling ) Quadratic Modeling ) Eponential Modeling Each tpe of modeling in mathematics is determined b the graph of equation for each model. In the net eamples, there is a sample graph of each tpe of modeling Linear models are described b the following general graph Quadratic models are described b the following general graph
Eponential models are described b the following general graph
Section 4.1 Linear Models Before ou can stud linear models, ou must understand so basic concepts in Algebra. One of the main algebra concepts used in linear models is the slope-intercept equation of a line. The slope intercept equation is usuall epressed as follows: Standard linear model m + b m slope b Intercept In this equation the variable m represents the slope of the equation and the variable b represents the -intercept of the line. When studing linear models, ou must understand the concept of slope. Slope is usuall defined as rise over run or change in over change in. In general slope measures the rate in change. Thus, the idea of slope has man applications in mathematics including velocit, temperature change, pa rates, cost rates, and several other rates of change. Slope rise Slope run m 1 1 change in change in Basic Algebra Skills (Slope and -intercept) In net eamples, we will find the slope of a line given two points on the line. Eample 1 Find the slope between the points (1,) and (,) m 1 1 1 1 1
Eample Find the slope between the points (,) and (4,6) 4 6 1 1 m Slope and -intercept also can be found from the equation in slope-intercept, as shown in this net eample. Notice that the equation is written in slope-intercept form. Eample Find the slope and -intercept b m If the equation is not written in slope intercept form, it can be rearranged to slopeintercept form b solving the equation for. This procedure is shown in the net two eamples. Eample 4 Find the slope and -intercept 6 6 6 6 + + + + + + b m
Eample Find the slope and -intercept 1 1 1 1 + + + b m Eample 6 Graph the equation First construct a table using 4 arbitrar values of, and then substitute these values to the equation to get the corresponding values. 1 1 (1) 1 () 9 () 4 4 6 (4)
Net make point using the four points in the above table. 4 - - -4-6 Applications of Linear Equations Eample 6 (Temperature conversion) F 9 C + 9 a) Sketch a graph of F C + C 1 4 9 F C + 9 F (1) + 9() + 18 + 9 F () + 9(4) + 6 + 68 9 F () + 9(6) + 4 + 86 9 F (4) + 9(8) + 7 + 14
b) Use the model to convert 1 degrees Celsius to degrees Fahrenheit. F F F F 9 C + 9 (1) + 16 + 48 c) Use the model to convert 1 degrees Fahrenheit to Celsius. 9 F C + 9 1 C + 9 1 C + 9 18 C 9 (18) C 9 9 C 1 C
Eample 7 (Business Applications) The revenue of a compan that makes backpacks is given b the formula where represents the number of backpacks sold. R 1. a) Graph the linear model R 1. X R 1. 1 R 1.(1) 1 R 1.() 4 R 1.() 64 4 R 1.(4) 86 b) Use the model to calculate the revenue for selling backpacks R 1. 1.() $17. c) What is the slope m $1. d) What is the meaning of the slope? Cost per unit sold Revenue made per backpack solid
Eample 8 (Sales) A salesperson is paid $1 plus $6 per sale each week. The model S 6 + 1 is used to calculate the salesperson s weekl salar where is the number of sales per week. a) Graph S 6 + 1 X S S 6 ( ) + 11 + 1 4 S 6 (4) + 1 4 + 1 4 6 S 6 (6) + 1 6 + 1 46 8 S 6 (8) + 1 48 + 1 8 b) Use the model to calculate the salespersons weekl salar if he/she makes 8 sales. S 6 (8) + 1 48 + 1 $8. c) What is the slope of the equation m 6 $ sale d) What is the meaning of the slope Dollars per each sale
Eample 9 Given the following data sketch a graph Time Temperature 1 min C min 7 C min 11 C 4 min 14 C Sketch a graph of the given data and then compute the slope of the resulting line. 1 1 8 (,7) 6 4 (1,) - 1 1 - Use the points (1,) and (,7) in the above graph to compute the slope 7 m 1 4 1 4
Eample 1 An approimate linear model that gives the remaining distance, in miles, a plane must travel from Los Angeles to Paris given b d 6 t where d is the remaining distance and t is the hours after the flight begins. Find the remaining distance to Paris after hours and hours. d d d d d d 6 () 6 16 4miles 6 () 6 7 miles How long should it take for the plane to flight from Los Angeles to Paris? 6 t + t 6 t + t t 6 t 6 t 1.9 hours
Problem Set 1 1) Find the slope between the points (1,1) and (,) ) Find the slope between the points (,) and (4,) Given the equation, find the slope and -intercept. ) 4 4) + 4 6 ) 6 Graph the following equations 6) 7) + 1 8) 1 4 9) 6 Linear Models 1) The revnue of a compan that makes backpacks is given b the formula R 4. where represents the number of backpacks sold. a) Graph the linear model R 4. b) Use the model to calculate the revenue for selling 4 backpacks? c) What is the slope of the model? d) What is the meaning of the slope? 11) A salesperson is paid $1 plus $ per sale each week. The model S + 1 is used to calculate the salesperson s weekl salar where is the number of sales per week. a) Graph S + 1 b) Use the model to calculate the salespersons weekl salar if he/she makes 8 sales. c) What is the slope of the equation? d) What is the meaning of the slope?
1) A salesperson is paid $ plus $ per sale each week. The model S + is used to calculate the salesperson s weekl salar where is the number of sales per week. a) Graph S + b) Use the model to calculate the salespersons weekl salar if he/she makes 8 sales. c) What is the slope of the equation? d) What is the meaning of the slope? 1) An approimate linear model that gives the remaining distance, in miles, a plane must travel from San Francisco to London given b d( t) t where d(t) is the remaining distance and t is the hours after the flight begins. Find the remaining distance to London after hours and 4 hours.
Section 4. Quadratic Models Graph of Quadratic Models The graph of a quadratic model alwas results in a parabola. The general form of a quadratic function is given in the following definition. A quadratic function is a function where the graph is a parabola and the equation is of the form: a + b + c where a b The -coordinate of verte is given b the equation: a The verte is the turning point on the graph of a parabola. If the parabola opens upward, then the verte is the lowest point of the graph. If the parabola opens downward, then the verte is the highest point on the graph. The direction of the parabola opens can be determined b the sign of the term or the a term in the above equation. If a <, then the parabola open downward. Similarl if a >, then the parabola opens upward. (See graphs below in figure 1-1) Figure 1-1 A parabola where a > and the verte is the lowest point on the graph
A parabola where a < and the verte is the highest point on the graph Here are some eamples of finding the verte and -intercepts of an eponential equation. The graph of the quadratic equation is also provided in these eamples Eample 1 Find the verte and -intercepts of the quadratic equation, and then make a sketch of the parabola. a 1, c (1) -intercepts: (,) and (,) Graph for Eample 1
Eample Find the verte and -intercepts of the quadratic equation, and then make a sketch of the parabola. 4 9 9 4 9 (1) Verte -intercepts (,) (,) ) ( and or
Graph of the function Eample Find the verte and -intercepts of the quadratic equation, and then make a sketch of the parabola. 6 Verte ( 6) () 1 (1, ) () 6() 1 -intercepts 6 6 1 6
6 ( ) or (,) and (,) Graph of 6 More about Quadratic Equations In some instances, the quadratic equation will not factor properl. In this case, ou must use what is called the quadratic formula. In the net few eamples, the quadratic formula will be used to find the solutions of a quadratic equation. The Quadratic Formula The solution to the equation a + b + c is given b b ± b 4ac a
Eample 4 Solve + 7 a 1 b c 7 b ± b 4ac a ± 4(1)( 7) (1) ± + 8 ± Eample Solve + 7 9 7 ± 7 4(1)( 9) (7) 7 ± 49 + 6 (7) 7 ± 14 8 Eample 6 At a local frog jumping contest. Rivet s jump can be approimated b the equation 1 + and Croak s jump can be approimate b 1 + 4, where the 6 length of jump in feet and the height of the jump in feet. a) Which frog can jump higher Rivet s verte: 6 Height: 1 (6) + (6) 6 + 1 6 ft 1 1 6 6 4 4 Croak s verte: 4 Height: 1 (4) + 4(4) 8 + 16 8 ft 1 1 Croak can jump higher at 8 feet b) Which frog can jump farther Rivet s can jump farther at (6 ft) 1 feet
Graph of the frogs jumps 8-1 g () ( ) +4 6 4-1 f () ( 6 ) + - - Using the parabola to find the maimum or minimum value of a quadratic function The parabola can be used to find either the maimum value or the minimum value of a quadratic function. (See figure 1-1) This can simpl be done b find the verte of the parabola. Remember as stated earlier the verte will turn out to be either the highest point on the curve or the lowest point on the curve. In the net eamples, the verte of the parabola will be use to find the maimum value. Eample 7 The path of a ball thrown b a bo is given b the equation.4 + 1. where is the horizontal distance the ball travels and is the height of the ball. Find the maimum height of the ball in ards. Find the verte of the ball 1. (.4) 1..8 18.7.4 18.7 ( ) + 1.(18.7) 14.1+ 8.1 14 ards
Eample 8 The path of a cannon ball is given b the equation.1 + 6. where is the horizontal distance the ball travels and is the height of the cannon ball. Find the maimum height of the cannon ball in feet. Find the verte of the cannon ball. 6. 6..1 9 (.1). ( ) + 6() 9 + 18 feet
Problem Set Find the verte and -intercepts of the given parabola, and then make a sketch of the parabola. 1) 4 ) 4 ) + 1 4) 4 + ) 16 6) 6 Quadratic Models 7) The path of a ball thrown b a baseball plaer is given b the equation. + 1. 6 where is the horizontal distance the ball travels and is the height of the ball. Find the maimum height of the ball in ards. 8) The path of a ball thrown b a bo is given b the equation.6 + 1. 8 where is the horizontal distance the ball travels and is the height of the ball. Find the maimum height of the ball in ards. 9) The path of a cannon ball is given b the equation. + 6. where is the horizontal distance the ball travels and is the height of the cannon ball. Find the maimum height of the cannon ball in feet. 11) The path of a cannon ball is given b the equation.1 + 8. where is the horizontal distance the ball travels and is the height of the cannon ball. Find the maimum height of the cannon ball in feet.
Section 4. Eponential models The eponential function e.718 The Euler number Eample 1 Simplif the following eponential functions 1) e ) e ) e 1 7.9 1 e 1.4. The graph of the eponential function Eample Graph e X Y - e. 14-1 e 1. 7 e o 1 1 e 1. 7 e 7. 4
Eample Graph 1e. -.( ) 1e.4 1e 6. 7-1.( 1) 1e. 1e 8..() 1e 1e 1 1.(1) 1e. 1e 1..() 1e.4 1e 14. 9
Eponential Models Eponential models are used to predict human populations, animal populations, mone growth, pollution growth, and other aspects of societ that fit eponential models. The variable of an eponential model is found in the eponent of the equation. Eponential Growth t P P (1 + r) P New Value P Original Value r rate t time
Eample 4 The population of the United States is 9 million, what would be the population of the U. S. be in ears if its population would growth at a stead rate of.7 % for ears? P P (1 + r) P 9,, r.7%.7 t P 9(1 +.7) t 9(1.7) 416746 Eample The population of Blacksburg, Virginia is 41,, what would be the population in 1 ears if Blacksburg would grow at a rate of 1.1 % per ear? P P (1 + r) P 41 r 1.1%.11 t 1 P 41(1 +.11) t 1 41(1.11) 1 474 Eample 6 In 199 the United States had greenhouse emissions of about 14 million tons, where as China had greenhouse emissions of about 8 million tons. If in the net ears China greenhouse emission grew b 4 percent and the U. S. greenhouse emission grew b 1. percent, what would the emissions in tons for both countries in?
U. S. Emissions in P P (1 + r) P 14 million r 1.%.1 t P 14(1 +.1) P P (1 + r) P 8 million r 4.%.4 t P 8(1 +.4) t China' s Emissions in t 14(1.1) 8(1.4) 19 million tons 6 million tons Eample 7 Using the eponential growth formula, find the amount of mone that ou would have in a bank account if ou deposited $, in the account for 1 ears at 1.1 % interest rate? P P (1 + r) P r 1.1%.11 t 1 P (1 +.11) t 1 (1.11) 1 $48.91 Eponential deca Eponential deca models are use to measure radioactive deca, decreasing populations, Half-life, and other elements that fit an eponential model. Again, the one variable in an eponential deca models in found in the eponent. Eponential Deca Formula
P P (1 r P New Value P Original Value r rate t time t ) Eample 8 A certain population of black bears in the eastern United States has been decreasing b.1 percent per ear. If this trend keeps up, what will be the population of bears in ears if there are currentl 1 bears. P P (1 r) P 1 r.1%.1 t P 1(1.1) t 1(.969) Eample 9 A certain isotope decreases at a rate of % per ear. It there is currentl 4 grams of the isotope, how man grams of the isotope will there be in ears? P P (1 r) P 4 r %. t P 4(1.) t 4(.9) 1 grams Problem Set Eponential Functions Evaluate using a calculator 1) e
1 ) e 4 ) e Graph the following functions 4) ) e 1 6) e 7) e Growth Models (Show Work) 8) The current population of German is 8,,. What would be the population of German in 1 ears if its population would growth at a stead rate of.9 % for 1 ears? 9) The current population of Salem, Virginia is,. What would be the population of Salem in ears if Salem would grow at a rate of 1. % per ear? 1) Using the eponential growth formula, find the amount of mone that ou would have in a bank account if ou deposited $1, in the account for 1 ears at 1.6 % interest rate?.t 11) A certain rabbit population is modeled b the equation P e where t is the time in months. Use the model to predict the population after months. Deca Models 11) A certain population of Panda Bears in China has been decreasing b 1. percent per ear. If this trend keeps up, what will be the population of Panda Bears in 1 ears if there are currentl bears? 1) A certain isotope decreases at a rate of 4% per ear. It there is currentl grams of the isotope, how man grams of the isotope will there be in ears?
Section 4. Basic Logarithms Definition of a Logarithm log b a b a Eample 1 i) Write 4as a logarithmic epression. 4 log 4 ii) Write 4 6 as a logarithmic epression. 4 6 log 6 4 Eample i) Write log 4 16 as eponential epression. log 16 4 4 16 ii) Write log 1 1, 4 as an eponential epression. log 1, 4 1 4 1 1, Log base ten Another wa of writing log 1 1 is log 1. The wa we find the answer to log 1 is to ask the question of 1 raised to what power gives ou 1? Since we know that1 4 1, the answer is 4.
Eample i) Find log 1, Since1 1,, log 1, ii) Find log 1 Since1 1, log 1 Eample 4 Use a scientific calculator to evaluate the following logarithms i) log 67 Answer: log 67.74 ii) log 89 Answer: log 89 4.49 iii) log 4678 Answer: log 4678.66 Graph of basic logarithms Eample Graph log6 log( 6()) log(1) 1. 7 1 log( 6(1)) log(6) 1. 8 log( 6()) log(1). 1 4 log( 6(4)) log(4). 4
Plot the given values from the table gives the following graph Eample 6 Graph log( + 1) log( + 1) log(). 4 1 log(1 + 1) log(11). log( + 1) log(1) 6. 6 4 log(4 + 1) log(41) 8. 1 Plot the given values from the table gives the following graph
Eample 7 (Using logarithmic models to model height) A logarithmic model to approimate the percentage P of an adult height a male has reached at an age A form 1 and 18 is P 16 log( A 1) + 84 1) Sketch a graph of this function. P A 1 P 16 log(1 1) + 84 84 14 P 16 log(14 1) + 84 16 log() + 84 4.8 + 84 88. 8 1 P 16 log(1 1) + 8 16 log() + 84 7.6 + 84 9. 6 18 P 16 log(18 1) + 84 16 log(6) + 84 1. + 84 96. Plot the given values from the table gives the following graph ) What does the graph tell ou about the height of male after age of 18? Usuall males stop growing after age 18 ) Use the model to compute the average height of a 16 ear old male. P 16 log(16 1) + 84 16.log(4) + 84 9.6 + 84 9.6 9.6%
Eample 8 Use the following model for $1 invested in saving account given b n 694. + 1.4 log( A), to find the amount of time (n) for the amount of mone A to grow to $1,. n 694. + 1.4 log1 n 694. + 1.4() n 694. + 117 n 46.8