#set document(title: "5.6 Projects for Chapter 5", author: "OpenStax / XYZ Homework") #set page(width: 8.5in, height: auto, margin: 1in) #import "@preview/cetz:0.5.2" #set text(font: ("STIX Two Text", "Libertinus Serif", "New Computer Modern"), size: 10.5pt, lang: "en") #show math.equation: set text(font: ("STIX Two Math", "New Computer Modern Math")) #set par(justify: true, leading: 0.62em, spacing: 0.9em) #set enum(spacing: 1.1em) // room between list items so tall inline fractions don't collide #set list(spacing: 1.1em) #set table(stroke: 0.5pt + rgb("#c7ccd3")) #let BLUE = rgb("#183B6F") // brand navy — section bars + example/solution labels (white on navy 11.09:1) #let ORANGE = rgb("#A94509") // brand primary-700 — AA-safe deep orange for TEXT (5.93:1 on white; raw brand #F37021 is 2.94:1 and must never carry text) #let RED = rgb("#DC2626") // brand error-600 #let GREEN = rgb("#059669") // brand success-600 (decoration only; small green text uses green-text #007942) #show heading.where(level: 1): it => block(width: 100%, above: 0pt, below: 16pt, fill: gradient.linear(BLUE, rgb("#2C5AA0")), inset: (x: 14pt, y: 12pt), radius: 3pt, text(fill: white, weight: "bold", size: 19pt, it.body)) #show heading.where(level: 2): it => block(width: 100%, above: 18pt, below: 10pt, fill: BLUE, inset: (x: 10pt, y: 6pt), radius: 2pt, text(fill: white, weight: "bold", size: 12pt, it.body)) #show heading.where(level: 3): it => text(fill: ORANGE, weight: "bold", size: 12.5pt, it.body) #show heading.where(level: 4): it => text(fill: BLUE, weight: "bold", size: 10.5pt, it.body) #let examplebox(label, title, body) = block(width: 100%, breakable: true, fill: rgb("#EFF1F5"), stroke: 0.5pt + rgb("#CFDDF0"), radius: 4pt, inset: 10pt, above: 12pt, below: 12pt)[ #block(below: 6pt)[#box(fill: BLUE, inset: (x: 6pt, y: 2pt), radius: 2pt, text(fill: white, weight: "bold", size: 8.5pt, label)) #h(0.4em) #strong[#title]] #body] // rail = decorative left rule (raw brand token); labelcolor = AA-safe label text shade #let notebox(label, rail, labelcolor, tint, body) = block(width: 100%, breakable: true, fill: tint, stroke: (left: 3pt + rail), inset: (left: 10pt, rest: 8pt), radius: (right: 4pt), above: 11pt, below: 11pt)[ #text(fill: labelcolor, weight: "bold", size: 7.5pt, tracking: 0.5pt)[#upper(label)] #linebreak() #body] #let solutionbox(body) = block(above: 4pt, below: 8pt)[ #text(fill: BLUE, weight: "bold", size: 8.5pt)[Solution] #linebreak() #body] #let figph(msg) = block(width: 100%, height: 60pt, fill: rgb("#f6f7f9"), stroke: (paint: rgb("#c7ccd3"), dash: "dashed"), radius: 4pt, inset: 10pt)[ #align(center + horizon, text(fill: rgb("#889"), style: "italic", size: 9pt, msg))] // Standardize inlined figure sizes: measure the natural CeTZ canvas, then scale to a // consistent envelope (aspect-aware; see build_typst.py FIG_* constants). Unlike the // print preamble, dimensions are FLOORED: in an editor a user can trim a figure to a // degenerate 1-D shape (a bare line), and w/h or tw/w would then divide by zero. #let _STD_W = 3.5 #let _WIDE_W = 5.6 #let _MAX_H = 3.4 #let _ASPECT_WIDE = 2.2 #let _UPSCALE_MAX = 1.15 #let stdfig(body) = context { let m = measure(body) let w = calc.max(m.width / 1in, 0.01) let h = calc.max(m.height / 1in, 0.01) let tw = if w / h > _ASPECT_WIDE { _WIDE_W } else { _STD_W } let s = calc.min(tw / w, _MAX_H / h, _UPSCALE_MAX) align(center, box(scale(x: s * 100%, y: s * 100%, reflow: true, body))) } #show figure: set block(breakable: false) #set figure(gap: 8pt) #show figure.caption: set text(size: 8.5pt, fill: rgb("#555")) == 5.6#h(0.6em)Projects for Chapter 5 #notebox("Project", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ #emph[The Logistic Function] In this project, we investigate the graph of the logistic function. + Graph the #strong[sigmoid function], #math.equation(block: false, alt: "s open parenthesis t close parenthesis equals the fraction 1 over 1 plus e to the power minus t")[$s ( t ) = display(frac(1, 1 + e^(− t)))$], in the window #math.equation(block: true, alt: "Xmin, equals minus 4, Xmax equals 4; Ymin, equals minus 1, Ymax equals 2")[$"Xmin" & = − 4 & & "Xmax" = 4 \ "Ymin" & = − 1 & & "Ymax" = 2$] What are the domain and range of the function? List the intercepts of the graph, as well as any horizontal or vertical asymptotes. Estimate the coordinates of the #strong[inflection point], where the graph changes concavity. + Graph the two functions #math.equation(block: false, alt: "Y sub 1 open parenthesis t close parenthesis equals the fraction 5 over 1 plus 4 e to the power minus t")[$Y_(1) ( t ) = display(frac(5, 1 + 4 e^(− t)))$] and #math.equation(block: false, alt: "Y sub 2 equals the fraction 10 over 1 plus 9 e to the power minus t")[$Y_(2) = display(frac(10, 1 + 9 e^(− t)))$] in the window #math.equation(block: true, alt: "Xmin, equals minus 2, Xmax equals 10; Ymin, equals minus 1, Ymax equals 11")[$"Xmin" & = − 2 & & "Xmax" = 10 \ "Ymin" & = − 1 & & "Ymax" = 11$] How do the graphs of these functions differ from the sigmoid function? State the domain and range, intercepts, and asymptotes of #math.equation(block: false, alt: "Y sub 1")[$Y_(1)$] and #math.equation(block: false, alt: "Y sub 2")[$Y_(2)$]. Estimate the coordinates of their inflection points. + The function #math.equation(block: false, alt: "P open parenthesis t close parenthesis equals the fraction K P sub 0 over P sub 0 plus open parenthesis K minus P sub 0 close parenthesis e to the power minus r t")[$P ( t ) = display(frac(K P_(0), P_(0) + ( K − P_(0) ) e^(− r t)))$] is called a #strong[logistic function]. It is used to model population growth, among other things. It has three parameters, #math.equation(block: false, alt: "K")[$K$], #math.equation(block: false, alt: "P sub 0")[$P_(0)$], and #math.equation(block: false, alt: "r")[$r$]. The parameter #math.equation(block: false, alt: "K")[$K$] is called the #strong[carrying capacity]. The functions #math.equation(block: false, alt: "Y sub 1")[$Y_(1)$] and #math.equation(block: false, alt: "Y sub 2")[$Y_(2)$] in part (b) are logistic functions with #math.equation(block: false, alt: "P sub 0 equals 1")[$P_(0) = 1$] and #math.equation(block: false, alt: "r equals 1")[$r = 1$]. What does the value of #math.equation(block: false, alt: "K")[$K$] tell you about the graph? What do you notice about the vertical coordinate of the inflection point? + Graph the function #math.equation(block: false, alt: "P open parenthesis t close parenthesis equals the fraction 10 P sub 0 over P sub 0 plus open parenthesis 10 minus P sub 0 close parenthesis e to the power minus t")[$P ( t ) = display(frac(10 P_(0), P_(0) + ( 10 − P_(0) ) e^(− t)))$] for #math.equation(block: false, alt: "P sub 0 equals 3")[$P_(0) = 3$], #math.equation(block: false, alt: "4")[$4$], and #math.equation(block: false, alt: "5")[$5$]. What does the value of #math.equation(block: false, alt: "P sub 0")[$P_(0)$] tell you about the graph? + Graph the function #math.equation(block: false, alt: "P open parenthesis t close parenthesis equals the fraction 20 over 2 plus 8 e to the power minus r t")[$P ( t ) = display(frac(20, 2 + 8 e^(− r t)))$] for #math.equation(block: false, alt: "r equals 0.5")[$r = 0.5$], #math.equation(block: false, alt: "1")[$1$], and #math.equation(block: false, alt: "2")[$2$]. What does the value of #math.equation(block: false, alt: "r")[$r$] tell you about the graph? ] #notebox("Project", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ #emph[Bell-shaped Curve] In this project, we investigate the normal or bell-shaped curve. + Graph the function #math.equation(block: false, alt: "f open parenthesis x close parenthesis equals e to the power minus x squared")[$f ( x ) = e^(− x^(2))$], in the window #math.equation(block: true, alt: "Xmin, equals minus 2, Xmax equals 2; Ymin, equals minus 1, Ymax equals 2")[$"Xmin" & = − 2 & & "Xmax" = 2 \ "Ymin" & = − 1 & & "Ymax" = 2$] What are the domain and range of the function? List the intercepts of the graph, as well as any horizontal or vertical asymptotes. Estimate the coordinates of the #strong[inflection point], where the graph changes concavity. + Graph the function #math.equation(block: false, alt: "f open parenthesis x close parenthesis equals e to the power minus open parenthesis x minus m close parenthesis squared")[$f ( x ) = e^(− ( x − m )^(2))$] for #math.equation(block: false, alt: "m equals minus 1")[$m = − 1$], #math.equation(block: false, alt: "0")[$0$], #math.equation(block: false, alt: "1")[$1$], and #math.equation(block: false, alt: "2")[$2$]. How does the value of #math.equation(block: false, alt: "m")[$m$] affect the graph? + The function #math.equation(block: true, alt: "N open parenthesis x close parenthesis equals the fraction 1 over s the square root of 2 π e to the power minus open parenthesis x minus m close parenthesis squared / 2 s squared")[$N ( x ) = display(frac(1, s sqrt(2 π))) e^(− ( x − m )^(2) / 2 s^(2))$] is called the #strong[normal] curve. It is used in statistics to describe the distribution of a variable, such as height, among a population. The parameter #math.equation(block: false, alt: "m")[$m$] gives the #strong[mean] of the distribution, and #math.equation(block: false, alt: "s")[$s$] gives the #strong[standard deviation]. For example, the distribution of height among American women has a mean of #math.equation(block: false, alt: "64")[$64$] inches and a standard deviation of #math.equation(block: false, alt: "2.5")[$2.5$] inches. Graph #math.equation(block: false, alt: "N open parenthesis x close parenthesis")[$N ( x )$] for these values + Graph the function #math.equation(block: true, alt: "N open parenthesis x close parenthesis equals the fraction 1 over s the square root of 2 π e to the power minus open parenthesis x minus m close parenthesis squared / 2 s squared")[$N ( x ) = display(frac(1, s sqrt(2 π))) e^(− ( x − m )^(2) / 2 s^(2))$] for #math.equation(block: false, alt: "s equals 0.5")[$s = 0.5$], #math.equation(block: false, alt: "0.8")[$0.8$], #math.equation(block: false, alt: "1")[$1$], and #math.equation(block: false, alt: "1.2")[$1.2$]. (You may have to adjust the window to get a good graph.) How does the value of #math.equation(block: false, alt: "s")[$s$] affect the graph? ] #notebox("Project", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ #emph[Hedgerows] Do hedgerows planted at the boundaries of a field have a good or bad effect on crop yields? Hedges provide some shelter for the crops and retain moisture, but they may compete for nutrients or create too much shade. Results of studies on the microclimates produced by hedges are summarized in the figure, which shows how crop yields increase or decrease as a function of distance from the hedgerow. (Source: Briggs, David, and Courtney, 1985) #figure(figph[crop yield near hedgerow], alt: "crop yield near hedgerow", caption: none) + We will use trial-and-improvement to fit a curve to the graph. First, graph #math.equation(block: false, alt: "y sub 1 equals x e to the power minus x")[$y_(1) = x e^(− x)$] in the window #math.equation(block: false, alt: "Xmin equals minus 2")[$"Xmin" = − 2$], #math.equation(block: false, alt: "Xmax equals 5")[$"Xmax" = 5$], #math.equation(block: false, alt: "Ymin equals minus 1")[$"Ymin" = − 1$], #math.equation(block: false, alt: "Ymax equals 1")[$"Ymax" = 1$] to see that it has the right shape. + Graph #math.equation(block: false, alt: "y sub 2 equals open parenthesis x minus 2 close parenthesis e to the power minus open parenthesis x minus 2 close parenthesis")[$y_(2) = ( x − 2 ) e^(− ( x − 2 ))$] on the same axes. How is the graph of #math.equation(block: false, alt: "y sub 2")[$y_(2)$] different from the graph of #math.equation(block: false, alt: "y sub 1")[$y_(1)$]? + Next we'll find the correct scale by trying functions of the form #math.equation(block: false, alt: "y equals a open parenthesis x minus 2 close parenthesis e to the power minus open parenthesis x minus 2 close parenthesis / b")[$y = a ( x − 2 ) e^(− ( x − 2 ) / b)$]. Experiment with different whole number values of #math.equation(block: false, alt: "a")[$a$] and #math.equation(block: false, alt: "b")[$b$]. How do the values of #math.equation(block: false, alt: "a")[$a$] and #math.equation(block: false, alt: "b")[$b$] affect the curve? + Graph #math.equation(block: false, alt: "y equals 5 open parenthesis x minus 2 close parenthesis e to the power minus open parenthesis x minus 2 close parenthesis / 4")[$y = 5 ( x − 2 ) e^(− ( x − 2 ) / 4)$] in the window #math.equation(block: false, alt: "Xmin equals minus 5")[$"Xmin" = − 5$], #math.equation(block: false, alt: "Xmax equals 25")[$"Xmax" = 25$], #math.equation(block: false, alt: "Ymin equals minus 20")[$"Ymin" = − 20$], #math.equation(block: false, alt: "Ymax equals 25")[$"Ymax" = 25$]. This function is a reasonable approximation for the curve in the figure. Compare the area of decreased yield (below the #math.equation(block: false, alt: "x")[$x$]-axis) with the area of increased yield (above the #math.equation(block: false, alt: "x")[$x$]-axis). Which area is larger? Is the overall effect of hedgerows on crop yield good or bad? + About how far from the hedgerow do the beneficial effects extend? If the average hedgerow is about #math.equation(block: false, alt: "2.5")[$2.5$] meters tall, how large should the field be to exploit their advantages? ] #notebox("Project", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ #emph[Carbon Content] Organic matter in the ground decomposes over time, and if the soil is cultivated properly, the fraction of its original organic carbon content is given by #math.equation(block: true, alt: "C open parenthesis t close parenthesis equals the fraction a over b minus the fraction a minus b over b e to the power minus b t")[$C ( t ) = display(frac(a, b)) − display(frac(a − b, b)) e^(− b t)$] where #math.equation(block: false, alt: "t")[$t$] is in years, and #math.equation(block: false, alt: "a")[$a$] and #math.equation(block: false, alt: "b")[$b$] are constants. (Source: Briggs, David, and Courtney, 1985) + Write and simplify the formula for #math.equation(block: false, alt: "C open parenthesis t close parenthesis")[$C ( t )$] if #math.equation(block: false, alt: "a equals 0.01")[$a = 0.01$], #math.equation(block: false, alt: "b equals 0.028")[$b = 0.028$]. + Graph #math.equation(block: false, alt: "C open parenthesis t close parenthesis")[$C ( t )$] in the window #math.equation(block: false, alt: "Xmin equals 0")[$"Xmin" = 0$], #math.equation(block: false, alt: "Xmax equals 200")[$"Xmax" = 200$], #math.equation(block: false, alt: "Ymin equals 0")[$"Ymin" = 0$], #math.equation(block: false, alt: "Ymax equals 1.5")[$"Ymax" = 1.5$]. + What value does #math.equation(block: false, alt: "C open parenthesis t close parenthesis")[$C ( t )$] approach as #math.equation(block: false, alt: "t")[$t$] increases? Compare this value to #math.equation(block: false, alt: "the fraction a over b")[$display(frac(a, b))$]. + The half-life of this function is the amount of time until #math.equation(block: false, alt: "C open parenthesis t close parenthesis")[$C ( t )$] declines halfway to its limiting value, #math.equation(block: false, alt: "the fraction a over b")[$display(frac(a, b))$]. What is the half-life? ] #notebox("Project", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ #emph[Change of Base] This project derives the #strong[change of base] formula. + Follow the steps below to calculate #math.equation(block: false, alt: "log sub 8 ⁡ open parenthesis 20 close parenthesis")[$log_(8) ( 20 )$]. - #strong[Step 1] #linebreak() Let #math.equation(block: false, alt: "x equals log sub 8 ⁡ open parenthesis 20 close parenthesis")[$x = log_(8) ( 20 )$]. Write the equation in exponential form. - #strong[Step 2] #linebreak() Take the logarithm base #math.equation(block: false, alt: "10")[$10$] of both sides of your new equation. - #strong[Step 3] #linebreak() Simplify and solve for #math.equation(block: false, alt: "x")[$x$]. + Follow the steps in part (a) to calculate #math.equation(block: false, alt: "log sub 8 ⁡ open parenthesis 5 close parenthesis")[$log_(8) ( 5 )$]. + Use part (a) to find a formula for calculating #math.equation(block: false, alt: "log sub 8 ⁡ open parenthesis Q close parenthesis")[$log_(8) ( Q )$], where #math.equation(block: false, alt: "Q")[$Q$] is any positive number. + Find a formula for calculating #math.equation(block: false, alt: "log sub b ⁡ open parenthesis Q close parenthesis")[$log_(b) ( Q )$], where #math.equation(block: false, alt: "b greater than 1")[$b > 1$] and #math.equation(block: false, alt: "Q")[$Q$] is any positive number. + Find a formula for calculating #math.equation(block: false, alt: "ln ⁡ open parenthesis Q close parenthesis")[$ln ( Q )$] in terms of #math.equation(block: false, alt: "log sub 10 ⁡ open parenthesis Q close parenthesis")[$log_(10) ( Q )$]. + Find a formula for calculating #math.equation(block: false, alt: "log sub 10 ⁡ open parenthesis Q close parenthesis")[$log_(10) ( Q )$] in terms of #math.equation(block: false, alt: "ln ⁡ open parenthesis Q close parenthesis")[$ln ( Q )$]. ] #notebox("Project", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ #emph[Log Equations] In this project, we solve logarithm equations with a graphing utility. We have already used the #strong[Intersect] feature to find approximate solutions for linear, exponential, and other types of equations in one variable. The same technique works for equations that involve common or natural logarithms. + Solve #math.equation(block: false, alt: "log sub 10 ⁡ open parenthesis x plus 1 close parenthesis plus log sub 10 ⁡ open parenthesis x minus 2 close parenthesis equals 1")[$log_(10) ( x + 1 ) + log_(10) ( x − 2 ) = 1$] using the #strong[Intersect] feature by setting #math.equation(block: false, alt: "Y sub 1 equals log ⁡ open parenthesis x plus 1 close parenthesis plus log ⁡ open parenthesis x minus 2 close parenthesis")[$Y_(1) = log ( x + 1 ) + log ( x − 2 )$] and #math.equation(block: false, alt: "Y sub 2 equals 1")[$Y_(2) = 1$]. What about logarithmic equations with other bases? The calculator typically does not have a log key for bases other than #math.equation(block: false, alt: "10")[$10$] or #math.equation(block: false, alt: "e")[$e$]. However, by using the change of base formula from Project 5, we can rewrite any logarithm in terms of a common or natural logarithm. + Use the change of base formula to write #math.equation(block: false, alt: "y equals log sub 2 ⁡ open parenthesis x close parenthesis")[$y = log_(2) ( x )$] and #math.equation(block: false, alt: "y equals log sub 2 ⁡ open parenthesis x minus 2 close parenthesis")[$y = log_(2) ( x − 2 )$] in terms of common logarithms. + Solve #math.equation(block: false, alt: "log sub 2 ⁡ x plus log sub 2 ⁡ open parenthesis x minus 2 close parenthesis equals 3")[$log_(2) x + log_(2) ( x − 2 ) = 3$] by using the #strong[Intersect] feature on your calculator. + Solve #math.equation(block: false, alt: "log sub 3 ⁡ open parenthesis x minus 2 close parenthesis minus log sub 3 ⁡ open parenthesis x plus 1 close parenthesis equals 3")[$log_(3) ( x − 2 ) − log_(3) ( x + 1 ) = 3$] by using the #strong[Intersect] feature on your calculator. ]