Gliding mammals Many species of small mammals (such as flying squirrels and marsupial gliders) have the ability to walk and glide. Recent research suggests that these animals choose the most energy-efficient means of travel. According to one empirical model, the energy required for a glider with body mass ?m? to walk a horizontal distance ?D? is 8.46 ?Dm?2?/?3(where ?m? is measured in grams, ?D? is measured in meters, and energy is measured in microliters of oxygen consumed in respiration). The energy cost of climbing to a height D tan ? ftand gliding at an angle o? ? (below the horizontal, with ??? = 0 representing perfectly horizontal flight and ?ft >? 45° representing controlled falling) a horizontal distance ?D? is modeled by 1.36 ?mD? tan ???. Therefore, the function ? gives the energy difference per horizontal meter traveled between walking and gliding: If ?S? > 0 for given values of ?m? and ??,? then it is more costly to walk than glide. a. For what glide angles is it more efficient for a 200 gram animal to rather than walk? b. Find the threshold function ??? = ?g?(?m?)that gives the curve along which walking and gliding are equally efficient. Is it an increasing or decreasing function of body mass? c. In order to make gliding more efficient than walking, do larger gliders have a larger or smaller selection of glide angles that they can use? d. Let ?ft? = 25° (a typical glide angle) and graph ?S? as a function ?of m? for 0 ? ?m? ? 3000 (in grams). For what values of m ? ? is gliding more efficient? e. For ??? =25°, what value of ?m? (call it ?m*?) maximizes S? f. Does ?m?*, as defined in part (e). increase or decrease with increasing ???? That is, as a glider reduces its glide angle, does its optimal size become larger or smaller? g. Assuming Dumbo is a gliding elephant whose weight is one metric ton (106 g), what glide angle would Dumbo use to be more efficient at gliding than walking? (?Source: Energetic savings and the body size distribution of gliding mammals,? Roman Dial, ?Evolutionary Ecology Research,? 5 (2003): 1151-1162.)

Solution Step 1 Given: the energy cost of climbing to a heightD tan and gliding at an angle ofa horizontal distanceDis modeled by1.36mD tan . Following function gives the energy difference per horizontal meter travelled between walking and gliding, Step 2 (a)Consider that the mass of the animal is, For the animal to glide rather than walk the energy required to walk must be greater than or equal to the energy required to glide, that is, S 0 In order to find the angle of glide,we need to solve S = 0 Therefore, Substitute m = 200in the above equation: 0 Hence, for the angles equal to = 46.78 , it is more efficient for a 200gmanimal to glide rather than walk. Step 3 (b)Consider the function This gives the energy difference per horizontal meter travelled between walking and gliding. The curve along which walking and gliding are equally efficient, is given by equating S = 0 That is, = tan (6.22m ) 31 Hence, the threshold function = g(m)that gives the curve along which walking and 1 31 gliding are equally efficient is given by = tan (6.22m ) It is a decreasing function of body mass.