61 according to the ideal gas law the pressure

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61. According to the ideal gas law, the pressure, temperature, and volume of a gas are related by P = kT / V , where k is a constant of proportionality. Suppose that V is measured in cubic inches (in 3 ), T is measured in kelvins (K), and that for a certain gas the constant of proportionality is k = 10 in · lb / K. (a) Find the instantaneous rate of change of pressure with respect to temperature if the temperature is 80 K and the volume remains fixed at 50 in 3 . (b) Find the instantaneous rate of change of volume with respect to pressure if the volume is 50 in 3 and the tem- perature remains fixed at 80 K. 62. The temperature at a point (x, y) on a metal plate in the xy -plane is T (x, y) = x 3 + 2 y 2 + x degrees Celsius. As- sume that distance is measured in centimeters and find the rate at which temperature changes with respect to distance if we start at the point ( 1 , 2 ) and move (a) to the right and parallel to the x -axis (b) upward and parallel to the y -axis. 63. The length, width, and height of a rectangular box are l = 5 , w = 2 , and h = 3, respectively. (a) Find the instantaneous rate of change of the volume of the box with respect to the length if w and h are held constant. (b) Find the instantaneous rate of change of the volume of the box with respect to the width if l and h are held constant. (c) Find the instantaneous rate of change of the volume of the box with respect to the height if l and w are held constant. 64. The area A of a triangle is given by A = 1 2 ab sin θ , where a and b are the lengths of two sides and θ is the angle between these sides. Suppose that a = 5, b = 10, and θ = π / 3. (a) Find the rate at which A changes with respect to a if b and θ are held constant. (b) Find the rate at which A changes with respect to θ if a and b are held constant. (c) Find the rate at which b changes with respect to a if A and θ are held constant. 65. The volume of a right circular cone of radius r and height h is V = 1 3 πr 2 h . Show that if the height remains constant while the radius changes, then the volume satisfies ∂V ∂r = 2 V r 66. Find parametric equations for the tangent line at ( 1 , 3 , 3 ) to the curve of intersection of the surface z = x 2 y and (a) the plane x = 1 (b) the plane y = 3 . 67. (a) By differentiating implicitly, find the slope of the hy- perboloid x 2 + y 2 z 2 = 1 in the x -direction at the points ( 3 , 4 , 2 6 ) and ( 3 , 4 , 2 6 ) . (b) Check the results in part (a) by solving for z and dif- ferentiating the resulting functions directly. 68. (a) By differentiating implicitly, find the slope of the hy- perboloid x 2 + y 2 z 2 = 1 in the y -direction at the points ( 3 , 4 , 2 6 ) and ( 3 , 4 , 2 6 ) . (b) Check the results in part (a) by solving for z and dif- ferentiating the resulting functions directly. 69–72 Calculate ∂z / ∂x and ∂z / ∂y using implicit differentia- tion. Leave your answers in terms of x , y , and z . 69. (x 2 + y 2 + z 2 ) 3 / 2 = 1 70. ln ( 2 x 2 + y z 3 ) = x 71. x 2 + z sin xyz = 0 72. e xy sinh z z 2 x + 1 = 0 73–76 Find ∂w / ∂x , ∂w / ∂y , and ∂w / ∂z using implicit differ- entiation. Leave your answers in terms of x , y , z , and w .

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