H(T,8.4) 500 408 H(T,8.3) 300 H(T,8,2) 200 100 H(T,8.1) 10 20 30 - -- -- t-k -L- J -J-L-L-L ---- - - -+ - -- -- - + - -L -I- J -J-L-L L-I-- -- -- -1-t -
An airport can be cleared of fog by heating the air. The amount of heat required depends on the air temperature and the wetness of the fog. The figure below shows the heat H(T,w) required (in calories per cubic meter of fog) as a function of the temperature T (in degrees Celsius) and the water content w (in grams per cubic meter of fog). Note that this figure is not a contour diagram, but shows cross-sections of H with w fixed at 0.1, 0.2, 0.3, and 0.4.
(a) Estimate HT (10,0.2): HT (10,0.2) ≈ (Be sure you can interpret this partial derivative in practical terms.)
(b) Make a table of values for H(T,w) from the figure, and use it to estimate HT (T,w) for each of the following: T = 20,w = 0.2 : HT (T,w) ≈ T = 30,w = 0.2 : HT (T,w) ≈ T = 20,w = 0.3 : HT (T,w) ≈ T = 30,w = 0.3 : HT (T,w) ≈
(c) Repeat (b) to find Hw(T,w) for each of the following: T = 20,w = 0.2 : Hw(T,w) ≈ T = 30,w = 0.2 : Hw(T,w) ≈ T = 20,w = 0.3 : Hw(T,w) ≈ T = 30,w = 0.3 : Hw(T,w) ≈ (Be sure you can interpret this partial derivative in practical terms.)
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