The average germination time g (measured in days) of a batch of seeds depends on

Question

The average germination time g (measured in days) of a batch of seeds depends on the temperature T (measured in degrees centigrade) and the water potential P (measured in megapascals). If the temperature is 30 degrees centigrade and the water potential is 1.5 megapascals, the germination time is 2.600 days. Values of the partial derivatives under these conditions are g T = 0.130 days per degree centigrade, and g P = 1.200 days per megapascal.

1) What would the approximate germination time be if the temperature was changed to 30.25 degrees centigrade, while keeping the water potential at 1.5 megapascals? (Give your answer correct to at least three decimal places.)______

2) What would the approximate germination time be if the water potential was changed to 1.2 megapascals, while keeping the temperature at 30 degrees centigrade? (Give your answer correct to at least three decimal places.)_______

3) What would the approximate germination time be if the temperature was changed to 30.25 degrees centigrade and the water potential was changed to 1.2 megapascals? (Give your answer correct to at least three decimal places.) ______

Explanation / Answer

g(30,-1.5)=2.6

g/T = 0.130 , g/P = 1.200

linear approximation, g(T,P)=g(30,-1.5)+(g/T)(T-30)+(g/P)(P-(-1.5))

g(T,P)=2.600 -0.130(T-30) -1.200(P+1.5)

(1)

the temperature was changed to 30.25 degrees centigrade, while keeping the water potential at 1.5 megapascals

approximate germination time,g(30.25,-1.5)=2.600 -0.130(30.25-30) -1.200(-1.5+1.5)

approximate germination time,g(30.25,-1.5)=2.600 -0.130(0.25) -0

approximate germination time,g(30.25,-1.5)=2.600 -0.0325

approximate germination time,g(30.25,-1.5)=2.5675

approximate germination time,g(30.25,-1.5)=2.568

(2)

the water potential was changed to 1.2 megapascals, while keeping the temperature at 30 degrees centigrade

approximate germination time,g(30,-1.2)=2.600 -0.130(30-30) -1.200(-1.2+1.5)

approximate germination time,g(30,-1.2)=2.600 -0 -1.200(0.3)

approximate germination time,g(30,-1.2)=2.600 -0.36

approximate germination time,g(30,-1.2)=2.240

(3)

temperature was changed to 30.25 degrees centigrade and the water potential was changed to 1.2 megapascals

approximate germination time,g(30.25,-1.2)=2.600 -0.130(30.25-30) -1.200(-1.2+1.5)

approximate germination time,g(30.25,-1.2)=2.600 -0.130(0.25) -1.200(0.3)

approximate germination time,g(30.25,-1.2)=2.600 -0.0325 -0.36

approximate germination time,g(30.25,-1.2)=2.2075

approximate germination time,g(30.25,-1.2)=2.208

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