Calculus I Section 4.7. Optimization Equation. Math 151 November 29, 2008
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1 Calculus I Section 4.7 Optimization Solutions Math 151 Novembe 9, 008 The following poblems ae maimum/minimum optimization poblems. They illustate one of the most impotant applications of the fist deivative. Many students find these poblems intimidating because they ae wod poblems, and because thee does not appea to be a patten to these poblems. Howeve, if you ae patient you can minimize you aniety and maimize you success with these poblems by following this stategy: STRATEGY FOR SOLVING MAX/MIN OPTIMIZATION PROBLEMS 1. Read each poblem slowly and caefully. Read the poblem at least thee times befoe tying to solve it. Sometimes wods can be ambiguous, so it is impeative to know eactly what the poblem is asking. If you misead the poblem o huy though it, you have NO chance of solving it coectly.. If appopiate, daw a sketch/diagam of the poblem to be solved. Pictues ae a geat help in oganizing and soting out you thoughts.. Define vaiables to be used and caefully label you pictue o diagam with these vaiables. This step is vey impotant because it leads diectly/indiectly to the ceation of mathematical equations. 4. Wite down all equations that ae elated to you poblem/diagam. Clealy denote the equation which you ae asked to maimize/minimize. Epeience will show you that MOST optimization poblems will begin with two equations. One equation is a constaint equation and the othe is the optimization equation. The constaint equation is used to solve fo one of the vaiables. Then this is substituted into the optimization equation befoe diffeentiation occus. Some poblems may have NO constaint equation. Some poblems may have two o moe constaint equations. 5. Befoe diffeentiating, make sue that the optimization equation is a function on only one vaiable. Then diffeentiate using the well-known ules of diffeentiation. 6. Veify that you esult is a maimum/minimum value by using the 1 st o nd deivative test fo etema. W Eecise 1: Find the dimensions that will maimize the total aea of a ectangula pen which is to be built with thee paallel patitions using 500 feet of fencing. Let L epesent the length in feet and W the width in feet. L Maimize Aea, A = LW Fom Peimete, L + 5W = 500 Optimization Equation Constaint Equation Solve fo W W = 100 5L sub. into Optimization Eqn. A(L = L ( L = 100L 5 L, 0 < L < 50 Find A (L = L = 0 L = 15 Since A (L = 4 5 < 0 fo all L Then L = 15 feet epesents the maimum. Maimum dimensions: Length 15 ft, Width 50 ft.
2 y 50t Eecise : An open ectangula bo with a squae base is constucted fom 48 squae feet of mateial. What dimensions will esult in a bo with the lagest volume? Let epesent the length (in ft. of one side of the base and h the height (in ft.. Volume, V = h, to Maimize ( optimization equation. Suface aea, A = h = 48 ( Constaint Equation h =, 0 < < 48. ( 4 48 The optimization equation becomes, V ( = = Setting V ( = = 0 = 16 = 4. Since V ( = V (4 = 6 Concave down fo = 4, veifying that at = 4 is a maimum. The maimum dimensions ae = 4 ft and h = ft. Eecise : A cylindical can is to hold $0 π m. The mateial fo the top and bottom costs $10 pe m and the mateial fo the sides costs $8 pe m. Find the adius,, and height, h, of the most economical can. Let be the adius (in m and h be the height (in m. The Aea of the cylinde, A = π + πh The Cost to make the can, C = 10(π + 8(πh = 0π + 16πh, (Equation to Minimize. The volume of the cylinde, π h = 0π (Constaint Equation h = 0π π = 0 The optimization equation becomes, C( = 0π + 16π ( 0 = 0π + 0π. Setting C ( = 0 40π 0π = 0 = 8 = Since C ( = 40π + 640π C ( = +, Making the function Concave up fo =, so a minimum eists at =. The minimum will eist when = m and h = 5 m. Eecise 4: Ca B is 0 miles diectly east of Ca A and begins moving west at 90 mph. At the same moment Ca A begins moving Noth at 60 mph. What will be the minimum distance between the cas and at what time t does the minimum distance occu? D(t = (60t + (0 90t ca A 0 90t 0 miles = 90t ca B Let t be the time in hous, then D(t = (60t + (0 90t is the equation to minimize. [ Finding D (t yields, D (t = 1 (60t + (0 90t ] 1/ {(60t(60 + (0 90t( 90} D 600t t (t = (60t + (0 90t = 11700t 700 (60t + (0 90t Want D (t = 0, ecall A 7 B = 0 A = 0, thus 11700t 700 = 0 t = hs. To veify this yields a minimum, use the check the value of the fist deivative on eithe side of.hs.
3 D (t. t D (.1 = D (1 = + The minimum occus at t =.hs with minimum distance will be D( miles. Eecise 5: A ectangula poste boad is to contain 108 cm of pinted mateial, with magins of 6 cm each at the top and bottom and cm on the sides. What is the minimal cost of the poste if it is to be pinted on mateial costing.0 $/cm. Let and y epesent the length and width of the pinted mateial espectively in cm. 6 y 6 Aea of the poste boad, A = ( + 4(y + 1 cm. Cost of the poste boad, C =.A =.( + 4(y + 1 $. This is the equation to minimize (optimization equation. Aea of pinted mateial is given as y = 108 (constaint equation. y = 108 Optimization equation becomes, C( =.( + 4 ( =. ( Setting C ( = 0 yields. ( 1 4 = 0 = 6 = 6. Fom C ( =. ( 864 and C (6 = +, so the function is concave up at = 6, thus a minimum cost occus when = 6 cm and y = 18 cm. fo the pinted mateial. Theefoe, the dimensions of the poste boad fo the minimum cost ae 10 cm by 0 cm.
4 Eecise 6: Find the dimensions of the cylinde of maimum volume that can be inscibed in a sphee of adius 1. Let and h be the adius and height of the inscibed cylinde. 1 h Maimize the Volume of the cylinde V = π h. We also know fom pythagoean theoem + ( h ( = 1, (constaint equation, = 1 h Optimization equation becomes, V (h = π 1 h 4 h = πh πh 4. Setting V (h = 0 yields π πh 4 = 0 h = 4 h =. ( Fom V (h = πh and V eists. The volume will be a maimum when h = and = =, the the function is concave down at h = and a maimum. 4.
5 Eecise 7: Two posts, one is 1 ft. high and the othe is 8 ft. high ae placed 0 ft. apat. They ae to be stayed by wies attached to a single stake unning fom gound level to the top of each post. Whee should the stake be placed to use the least wie? Let epesent the distance to place the stake fom the 1 foot post. D ( 8 1 D 1 ( 0 0 The distance to the stake D 1 ( = 1 + and D ( = (0 + 8, total distance of the wie, D( = D 1 ( + D ( = ( Find D ( = 0. ( D ( = / ( {} + 1 ( / {(0 ( 1} D ( = (0 + 8 D ( = = 0 (0 + 8 (0 + 8 = (0 1 + ( (0 + 8 = (0 ( 1 + (0 + 8 = (0 1 + (0 8 = (0 1 8 = = 60 = 9 Let s find the sign of the fist deivative on eithe side of = 9. D ( 9 D (7 = D (10 = + Thus to use the minimum amount of wie the state should be placed 9 feet fom the 1 foot post.
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