Opimizaion: finding he bigges/smalles/highes/lowes, ec. Los of non-sandard problems! Logisic growh rae 7.1 Simple biological opimizaion problems Small populaions AND large populaions grow slowly N: densiy of he populaion, G: growh rae Logisic growh law: ( ) K N G(N) = rn K r, K: consans Example 1: For which values of N is G(N) = 0? Wha does G(N) < 0 mean? Give an inerpreaion of he consan K in our model. Wha populaion densiy achieves maximum growh rae? Fencing a pen 7.2 Opimizaion wih a consrain wall fence Example 2: Given 60 meres of fencing maerial, wha is he larges recangular area you can enclose?
Fencing a pen 7.2 Opimizaion wih a consrain w l Soluion 2: Area is l w, where l is he lengh of he side parallel o he wall, and w is he lengh of he side perpendicular o he wall. The amoun of fencing we have is 60 meres, so l + 2w = 60. Then l = 60 2w So, our area (in erms of only one variable w) is A = (60 2w)w This is a parabola poining down. Is maximum occurs when w = 15. Then he area of he pen is (60 15)(15) = 675 sq m. Cell surface area 7.2 Opimizaion wih a consrain Some cells are roughly shaped like cylinders. Q: Wha are he dimensions of he cell? Fixed volume, V Minimal surface area Cell surface area: analyse he problem 7.2 Opimizaion wih a consrain V : volume (fixed consan) r: radius L: lengh L r Quesion: wha are L and r (in erms of V ) giving minimal surface area?
Cell surface area: plan a sraegy 7.2 Opimizaion wih a consrain Find L, r ha make he surface area of he cylinder minimal. Find global minimum (using criical poins, ec.) Find an equaion for he surface area of he cylinder Quesion: wha are L and r (in erms of V ) giving minimal surface area? Cell surface area: implemen sraegy 7.2 Opimizaion wih a consrain Task: find an equaion for he surface area of he cylinder. r r πr 2 2πrL L L 2πr πr 2 Surface Area = 2πr 2 + 2πrL Cell surface area: plan a sraegy 7.2 Opimizaion wih a consrain Find L, r ha make he surface area of he cylinder minimal. Find global minimum (using criical poins, ec.) Find an equaion for he surface area of he cylinder SA = 2πr 2 + 2πrL Problems: wo variables Haven used V Quesion: wha are L and r (in erms of V ) giving minimal surface area?
Cell surface area: plan a sraegy 7.2 Opimizaion wih a consrain Find L, r ha make he surface area of he cylinder minimal. Find global minimum (using criical poins, ec.) Simplify surface area equaion ino one variable Wrie L as a funcion of r Find an equaion for he surface area of he cylinder: SA = 2πr 2 + 2πrL Quesion: wha are L and r (in erms of V ) giving minimal surface area? Cell surface area: implemen sraegy 7.2 Opimizaion wih a consrain Task: wrie L as a funcion of r. r Given: he volume is a consan, V. L The volume of a cylinder is: (Area of base) (heigh) So, V = πr 2 L L = V πr 2 Cell surface area: plan a sraegy 7.2 Opimizaion wih a consrain Find L, r ha make he surface area of he cylinder minimal. Find global minimum (using criical poins, ec.) Simplify surface area equaion ino one variable Wrie L as a funcion of r L = V πr 2 Find an equaion for he surface area of he cylinder: SA = 2πr 2 + 2πrL SA = 2πr 2 + 2πr ( V πr 2 ) SA = 2πr 2 + 2V r 1 Quesion: wha are L and r (in erms of V ) giving minimal surface area?
Cell surface area: implemen sraegy 7.2 Opimizaion wih a consrain Task: find minimum value of he funcion SA(r) = 2πr 2 + 2V r 1, r > 0. SA (r) = 4πr 2V r 2 0 = 4πr 2V r 2 2V r 2 = 4πr 2V = 4πr 3 V 2π = r 3 r = 3 V 2π The only criical poin (wih posiive r) is he one above. We should check ha i s a minimum. We choose o use he second derivaive es. SA (r) = 4π + 4V r 3 Since his is posiive for all posiive r, our criical poin is indeed a minimum. Cell surface area: plan a sraegy 7.2 Opimizaion wih a consrain Find L, r ha make he surface area of he cylinder minimal. Simplify surface area equaion ino one variable Find an equaion for he surface area of he cylinder: SA = 2πr 2 + 2πrL r = 3 V 2π L =? 3 4V π Find global minimum (using criical poins, ec.) Wrie L as a funcion of r L = V πr 2 Quesion: wha are L and r (in erms of V ) giving minimal surface area? Cell surface area 7.2 Opimizaion wih a consrain Some cells are roughly shaped like cylinders. Fixed volume, V Minimal surface area Q: Wha are he dimensions of he cell? Radius: 3 Lengh: 3 V 2π 4V π
Kepler s Wedding 7.2 Opimizaion wih a consrain Kepler S wine seller comforable wih mah wans cheap wine wine cask lengh S deermines price 7.2 Opimizaion wih a consrain Kepler s Wedding: Choose he Bigges Cask Suppose S is he same in all hese casks. (Therefore, hey all have he same price.) Which has he bigges volume of wine? S S (A) (B) S S (C) (D) Kepler s Wedding 7.2 Opimizaion wih a consrain Suppose S = 1 mere. Wha are he heigh and radius of a (cylindrical) cask wih maximum volume? r 1 h
Kepler s Wedding 7.2 Opimizaion wih a consrain Opimal cask: Kepler s Wedding, Coninued 7.3 Checking endpoins When S = 1, we found: V = π ( 16 4h h 3 ) The only criical poin was a h = 2 3 0.67. Example 3: Suppose he casks all have heighs ha are beween 0.5 meres and 1 mere. Wha is heigh of he cask wih he leas amoun of wine? Opimal Foraging 7.4 Opimal foraging More foraging yields more food Animal mus commue o food pach
Opimal Foraging 7.4 Opimal foraging Each nu has a hard shell, and i akes ime o crack one a a ime before I can ea he nuy goodness inside. A firs, he bees aacked, and I go very lile honey. Over ime, hey ired, and eaing honey go easier and easier. Collecing food was going grea, unil he birds showed up and sared sealing away my sash. A B C D E F 7.4 Opimal foraging Opimal Foraging wih Muliple, Equivalen Paches Sraegy 1: always go o he pach wih he mos amoun of food Sraegy 2: say in a pach unil you ea everyhing Opimal Foraging 7.4 Opimal foraging The energy gained from minues a a pach is given by f () = where E and k are posiive consans. E k + A B C D E F
Opimal Foraging 7.4 Opimal foraging The energy gained from minues a a pach is given by f () = Inerpre E and k. E k +. A B C D E F Wha o Opimize 7.4 Opimal foraging Idea 1: opimize oal energy consumed. Drawback: Idea 2: R() = energy consumed ime spen 7.4 Opimal foraging Opimize efficiency of energy consumpion Energy gained afer hours in a pach: Time needed o ravel o a pach: τ. E k + energy consumed R() = ime spen E k+ = + τ = E (k + )(τ + ) Example 4: Find ha makes R() maximum. This is he opimal residence ime.
E (k+)(τ+). 7.4 Opimal foraging Le R() = We can make a skech. We see a single global max. Finding he criical poin: R () = (k + )(τ + )(E) E((k + ) + (τ + )) (k + ) 2 (τ + ) 2 = E(kτ 2 ) (k + ) 2 (τ + ) 2 0 = E(kτ 2 ) (k + ) 2 (τ + ) 2 2 = kτ = kτ We ignore he roo for < 0. More Pracice You have L meres of rope, and you wan o use i o form a circle and a square. How would you enclose he mos area? The leas? A recangle is inscribed in a semicircle of radius R so ha one side of he recangle lies along a diameer of he semicircle. Find he larges and smalles possible perimeer of such a recangle. More Pracice Find he minimum disance from he poin (a, 0) o he parabola y 2 = 8x. y a x Find a poin A on he posiive x-axis and a poin B on he posiive y-axis such ha (i) he riangle AOB conains he firs quadran porion of he parabola y = 1 x 2 and (ii) he area of he riangle AOB is as small as possible y y = 1 x 2 B O A x