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BioModels - Example(Body Surface Area The body's largest...

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Example: (Body Surface Area) The body’s largest organ is the skin. In addition to other things, it protects the organs of the body from injury and regulates temperature. Body Surface Area, BSA, is used by doctors to calculate drug dosages, ventilation, fluid requirements and for extracorporeal circulation. In emergent clinical situations, the following linear function can be used to quickly approximate BSA to within 0.8-2.5% of estimates given by the best known formulas. Let w be the weight (in kg) and BSA the body surface area (in m 2 ). Then BSA( w ) = 1 30 ( w + 4) , 3 w 30 . Since the domain is the set of all w satisfying 3 w 30 , this formula only applies only to children and adults 1 . Based on one interpreation of the slope of a line, an increase in weight by 1 kg results in an increase in body surface area of 1/30 of a square meter. Example: (Celcius-Fahrenheit scales) Let F denote degrees in Fahrenheit and C in degrees Celcuis. Then F is a linear function of C, and we can find the equation using the two points: temperature of water at freezing: (0 , 32) temperature of water at boiling: (100 , 212) Therefore, slope = Δ F Δ C = 212 - 32 100 - 0 = 180 100 = 1 . 8 . We conclude that F = 1 . 8 C + b, for some value of b. We can find b by substituting either point into this equation, and we find that F = 1 . 8 C + 32 = 5 9 C + 32 . Example: (Ideal Gas Law) For a fixed volume, V, the pressure, P, (in psi) of a gas is directly proportional to the temperature, T, (in Kelvin). Recall that x C = ( x + 273) K. If your tire pressure is 36 psi at 27 C, then what is your tire pressure at night when the temperature is only 10 C? Since P T, there is a constant m such that P = mT. Next, 27 C = (27 + 273) K = 300 K, and substituting this data point into P = mT gives 36 = 300 m, so m = 012 . Therefore, P ( T ) = 0 . 12 T and, since 10 C = (10 + 273) K = 283 K, we get P (383) = 0 . 12(283) = 33 . 96 psi. Suppose the temperature, T, of a gas is fixed. Then the Ideal Gas Law implies that pressure, P, and volume, V, are inversely proportional. Therefore, P = c/V for some constant c. Example: (Territorial Animal) An animal defends a circular region of radius r. Suppose that the energy lost, L, due to hunting and defending the territory, is proportional to the area, A, and that the energy gained, G, is proportional to the circumference, C.
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