PERFUME BRIEF ABSTRACT A number of attempts are given to make an aesthetically pleasing perfume bottle using integration using both numerical estimates and exact methods. GENERAL INFORMATION FileName: PERFUME Full title: Designing a Perfume Bottle Developer: Susan R. Clements, Department of Mathematics, South Vigo High School, 3737 S. 7th St., Terre Haute, IN 47802 Contact: Brian J. Winkel, Department of Mathematics, Rose-Hulman Institute of Technology, Terre Haute IN 47803 USA. Phone: 812-877-8412. Email: winkel@rose-hulman.edu. FAX: 812-877-3198. Support: The production of this material is supported by the National Science Foundation under Division of Undergraduate Education grant DUE-9352849: Development Site for Complex, Technology-Based Problems in Calculus with Applications in Science and Engineering and the Arvin Foundation of Columbus IN. STATEMENT OF PROBLEM A perfume designer has commissioned you to design an attractive perfume bottle for her newly created fragrance. She has set a few requirements for the project. a.) The bottle must hold 1 fluid ounce of perfume. b.) The base of the bottle is to be perpendicular to the vertical axis. c.) The cross sections are to be circular. d.) The sides of the bottle must be curved. Guideline to Solution of the Problem a.) Approximate the volume necessary to hold 1 ounce of perfume. b.) Consider the dimensions of a right square prism with this volume. c.) Now change the base of the prism from a square to a circular base. d.) To make the sides of the bottle curved, the radii of consecutive circular cross-sections would need to vary. Use your judgement and draw the side of the bottle on graph paper. The x-axis would represent the median height of the cross section, and the y-axis would represent the radius of that particular section. e.) Estimate the volume again using the sum of the volumes of the cross sections and fine tune the graph. f.) Record the ordered pairs constituting the curve and fit a polynomial to the curve. Plot the polynomial curve on the same graph as the ordered pairs. g.) Find the volume determined by revolving this polynomial curve about the x-axis. Consider the fill line of the bottle. How does the result compare to your estimate? h.) How does the average value of the function compare to the radius that you considered in part c? i.) How thick is the glass forming the bottle? What is the equation of the curve comprising the outside of the bottle? j.) Determine the volume of glass necessary to create the bottle. Consider the base and any extra glass at the top. KEYWORDS Volume, plotting, curve fitting, integration, derivative, optimization, surface area, aesthetic design. TEACHER NOTES ISSUES RELATED TO THE PROBLEM An objective of this problem is that the students will realize some of the concepts that may be considered in the design process. The student will move from estimation to point plotting to curve fitting to integration in an effort to ascertain the projected volume. Along the way, the student will need to adjust components to arrive at a good solution. The design process will be carried out a little further as the student is given the problem that economics plays a role in industry. Through design of the package, the shipping carton, and an estimate of the cost of the bottle, package, and carton (materials only), the student is forced to solve a problem that will take ingenuity as well as methods of calculus and finite mathematics. This problem-solving method may well be trial and error. Prerequisites Curve fitting to data, summation, integration, volumes. Time allotment - time management Expectations Future payoffs Extensions k.) The bottle will be a spray bottle. Experiment. When defining a spray as one squirt, what is the volume per spray of a typical perfume bottle? Create a function which will give the remaining height of perfume after "s" sprays. The height of remaining perfume will change the most rapidly after how many sprays? What is the typical number of sprays a customer will use per day? Survey perfume users as to this question and as to the frequency of use of a favorite brand of perfume. Using this information, how often will the consumer need to replace the bottle of perfume? Define the amount remaining per bottle as a function of days? How much is the consumer willing to pay for this size bottle of perfume? The experimentation is completely up to you. The volume of remaining perfume is a function of height (x). The height needs to be represented as a number of sprays. A composite function may be used to express volume as a function of "s" sprays. The most rapid change can then be found by using derivatives. As for the remainder of question (k), collect data from your peers and be able to back your answers to the remaining questions. l.) Design a package for the perfume bottle. What is the minimum surface area for this package? The container must be rigid. Would a rectangular prism, a cylinder, or some other design be the best? Justify your answer. m.) The bottles will be shipped in cartons of 36. What is the minimum amount of material necessary for the carton? If there is more than one layer, a divider needs to be included between layers. The carton must be capable of supporting the contents. n.) Suppose a shipping company charges by volume. Does this influence your carton? If so, how? o.) The desirable carton considering m and n would then be one with a minimum volume and a minimum surface area or a compromise between the two. What are the dimensions of the ideal carton? p.) If packaging material for the bottle costs .10 per square meter, material for the carton costs .15 per square meter, and glass costs .05 per cubic centimeter, what is the cost of a carton of perfume bottles, packages, and carton? References and Sources POSSIBLE SOLUTION(S) a.) Approximate the volume necessary to hold 1 ounce of perfume: Since 1 lb = 454 grams, 1 oz = 28.38 grams. Therefore, 1 ounce of perfume would have a capacity approximately the same as water and would be about 28.38 cubic centimeters. 454/16//N 28.375 b.) Consider the dimensions of a right square prism with this volume: A right square prism with this volume could have possible dimensions of 4 X 7.1 approximately or any number of possibilities. Remember, the bottle must be capable of holding slightly more than the 1 ounce of perfume. c.) Now change the base of the prism from a square to a circular base: A circle having an area of 4, the same area as the base of the right prism above, would have a radius Solve[Pi*r^2==4,r]//N {{r -> -1.12838}, {r -> 1.12838}} d.) To make the sides of the bottle curved, the radii of consecutive circular cross-sections would need to vary. Use your judgement and draw the side of the bottle on graph paper. The x-axis would represent the median height of the cross section, and the y-axis would represent the radius of that particular section. We offer up a sample shape with the following data. data={{0,.3}, {1,.5}, {2,.9},{3,1.0},{4,1.25},{5,1.35}, {6,1.1},{7,.9},{7.5,.2}}; bottleside = ListPlot[data, PlotStyle -> {Dashing[{.05,.05}]}, PlotJoined -> True, AxesOrigin ->{0,0}] e.) Estimate the volume again using the sum of the volumes of the cross sections and dine tune the graph. I am estimating the volume by using the y value of the midpoint of two consecutive data points as the radius of the base of a cylindrical section and the difference of the x values as the height of the cylinder. Cylinder # 1 (0,.3) to (1,.5) Pi((.3+.5)/2)^2 *(1-0)//N 0.502655 Cylinder #2 (1,.5) to (2,.9) Pi((.5+.9)/2)^2 * (2-1)//N 1.53938 Continuing in this manner, the volume is approximately________. Or we turn to a summation approach using the Sum command. OverAllEstimate = Sum[Pi*((data[[i]][[2]] + data[[i+1]][[2]])/2)^2* (data[[i+1]][[1]] - data[[i]][[1]]), {i,1, Length[data]-1}]//N 22.4938 f.) Record the ordered pairs constituting the curve and fit a polynomial to the curve. Plot the curve on the same graph as the ordered pairs. This is an example of fitting a 7th degree polynomial to the data points. A 7th degree polynomial may not be the best to use. Experiment. f[x_] = Fit[data,{1,x,x^2,x^3,x^4,x^5,x^6,x^7},x] 2 3 4 0.300107 - 1.69806 x + 3.96473 x - 2.95179 x + 1.06767 x - 5 6 7 0.201978 x + 0.0191808 x - 0.000722175 x fts = Plot[f[x],{x,0,7.4},AxesOrigin -> {0,0}] Show [bottleside,fts] g.) Find the volume determined by revolving this polynomial about the x-axis. Consider the fill line of the bottle. How does it compare to your estimate? NIntegrate[Pi * f[x]^2,{x,0,7.1}] 22.653 This compares favorably with our summation approach above and so we have several "reliable" approaches: summation and fitting a curve followed by integrating. Obviously, the perfume bottle will not hold the ounce of perfume....and it's quite ugly to boot. The polynomial also doesn't seem to fit the data too well, but this gives a little bit of the idea behind the problem. h.) How does the average value of the function compare to the radius that you considered in part c? avg = (1/(7.1-0))*NIntegrate[f[x],{x,0,7.1}] 0.944289 difference = 1.13-avg 0.185711 The average radius of the cross section is too small. Note that just because the average is correct does not necessarily mean that the volume will be correct. i.) How thick is the glass forming the bottle? What is the equation of the curve comprising the outside of the bottle? I have decided that the thickness of the glass shall be a uniform .3 cm. The equation of the ouside of the bottle will then be f[x] + .3. j.) Determine the volume of glass necessary to create the bottle. Consider the base and any extra glass at the top. The top of the bottle could be either of glass or another material. I am considering it to be of glass with a cylindrical opening of .1 cm. radius. As stated earlier, the bottom is .3 cm thick and all sides are also. Volume of top: bottletop = NIntegrate[Pi*(f[x]+.3)^2,{x,7.2,7.5}]-Pi*.1^2*.3//N 0.635045 Volume of bottom: bottlebottom = Pi *.3^3//N 0.084823 Volume of sides: sides = Pi * NIntegrate[(f[x]+.3)^2-f[x]^2,{x,0,7.2}]//N 14.8225 The sum of the volumes of the top, bottom, and sides is: sides + top + bottom 14.8225 + bottom + top ISSUES IN SOLUTION There is not one correct answer. The student should be able to support his answer. Mathematica has been used in the possible solution of the problem. All portions could be done with or without a computer algebra system. Being able to fit a polynomial to the set of projected data points would be the main concern.