ate ane. 2 n те d 5 S 5 at y n ot 1+ uvw 30. L=xze-²-² 29. Raß² cos y 31. If z = 5x² + y² and (x, y) changes from (1, 2) to (1.05, 2.1), compare the values of Az and dz. 32. If z = x² - xy + 3y2 and (x, y) changes from (3,-1) to (2.96,-0.95), compare the values of Az and dz. 33. The length and width of a rectangle are measured as 30 cm and 24 cm, respectively, with an error in measurement of at most 0.1 cm in each. Use differentials to estimate the maxi- mum error in the calculated area of the rectangle. 34. Use differentials to estimate the amount of metal in a closed cylindrical can that is 10 cm high and 4 cm in diameter if the metal in the top and bottom is 0.1 cm thick and the metal in the sides is 0.05 cm thick. 35. Use differentials to estimate the amount of tin in a closed tin can with diameter 8 cm and height 12 cm if the tin is 0.04 cm thick. 36. The wind-chill index is modeled by the function W = 13.12 + 0.6215T-11.370.16 + 0.3965Tv0.16 where T is the temperature (in °C) and v is the wind speed (in km/h). The wind speed is measured as 26 km/h, with a possible error of ±2 km/h, and the temperature is measured as -11°C, with a possible error of ±1°C. Use differentials to estimate the maximum error in the calculated value of W due to the measurement errors in T and v. 37. The tension T in the string of the yo-yo in the figure is mgR 2r² + R² T= where m is the mass of the yo-yo and g is acceleration due to gravity. Use differentials to estimate the change in the tension if R is increased from 3 cm to 3.1 cm and r is increased from 0.7 cm to 0.8 cm. Does the tension increase or decrease? H 38. The pressure, volume, and temperature of a mole of an ideal gas are related by the equation PV = 8.31T, where P is mea- sured in kilopascals, V in liters, and T in kelvins. Use differ-