85 926CHAPTER 14 Partial Derivatives80. If u = ex+₂x₂+...+an, where a² + a² + + az = 1,show thatð²µ ²+əx² əx²81. The diffusion equationc(x, t)əcat==+.+R R₁***= Dwhere D is a positive constant, describes the diffusion of heatthrough a solid, or the concentration of a pollutant at time t ata distance x from the source of the pollution, or the invasion ofalien species into a new habitat. Verify that the functiondPdL1√4TDt+0²cəx²อรินəx²²2²is a solution of the diffusion equation.82. The temperature at a point (x, y) on a flat metal plate is givenby 7(x, y) = 60/(1 + x² + y²), where T is measured in °Cand x, y in meters. Find the rate of change of temperature withrespect to distance at the point (2, 1) in (a) the x-direction and(b) the y-direction.-x²/(4Dt)83. The total resistance R produced by three conductors with resis-tances R₁, R₂, R3 connected in a parallel electrical circuit isgiven by the formulaR₂ R3P= αL..= ULӘРар+ K = (a + B)PƏLollo akFind ƏR/ƏR₁.84. Show that the Cobb-Douglas production function P = bL KBsatisfies the equation85. Show that the Cobb-Douglas production function satisfiesP(L, Ko) = C₁(Ko)La by solving the differential equationto (See Equation 6.)86. Cobb and Douglas used the equation P(L, K) = 1.01L0.75 0.25to model the American economy from 1899 to 1922, where Lis the amount of labor and K is the amount of capital. (SeeExample 14.1.3.)(a) Calculate PL and PK.(b) Find the marginal productivity of labor and the marginalproductivity of capital in the year 1920, when L = 194 andK = 407 (compared with the assigned values L = 100 andK = 100 in 1899). Interpret the results.00(c) In the year 1920, which would have benefited productionmore, an increase in capital investment or an increase inspending on labor?87. The van der Waals equation for n moles of a gas isn²a(P + a)(v1where P is the pressure, Vis the volume, and Tis the tempera-ture of the gas. The constant R is the universal gas constantand a and b are positive constants that are characteristic of aparticular gas. Calculate aT/OP and aP/av.sb adᎧᏢ ᎧᏙ ᎧᎢᏙ ᎧᎢ ᎧᏢ88. The gas law for a fixed mass m of an ideal gas at absolutetemperature T, pressure P, and volume V is PV = mRT, whereR is the gas constant. Show thatnb)=89. For the ideal gas of Exercise 88, show thatᎧᏢ ᏍᏙT-ƏT ƏT= = nRT=R=C-FmR08 (b)C-1/4-190. The wind-chill index is modeled by the functionW = 13.12 + 0.6215T - 11.37 0.16 + 0.3965Tv0.16where T is the temperature (°C) and v is the wind speed(km/h). When T = -15°C and v= 30 km/h, by how muchwould you expect the apparent temperature W to drop if theactual temperature decreases by 1°C? What if the wind speedincreases by 1 km/h?91. A model for the surface area of a human body is given by thefunctionS = f(w, h) =0.1091w0.425h0.725where w is the weight (in pounds), h is the height (in inches),and S is measured in square feet. Calculate and interpret the3001 200partial derivatives.as(a) (160, 70)δω92. One of Poiseuille's laws states that the resistance of blood flow-ing through an artery isas(160, 70)dhwhere L and r are the length and radius of the artery and Cisa positive constant determined by the viscosity of the blood.Calculate R/L and OR/or and interpret them.93. In the project on page 344 we expressed the power needed by abird during its flapping mode asP(v, x, m) = Av³ +Vwhere A and B are constants specific to a species of bird, visthe velocity of the bird, m is the mass of the bird, and x is thefraction of the flying time spent in flapping mode. Calculateap/av, aP/ax, and P/am and interpret them.B(mg/x)²

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