Applications of Exponential & Log Function Derivatives

6. A casting is taken from one oven at 1500F and placed into another at 0F, but rising at a rate of 100^{o}/hr. The temperature is then:

T = 100t + 1500

Find the minimum temperature and when it occurs.

You know the steps: Take the derivative with respect to time, zero it equal to zero, and solve.

dT/dt = 100 + 1500(-0.2)e^{-0.2t}

0 = 100 – 300e^{-0.2t}

100 = 300e^{-0.2t}

1/3 = e^{-0.2t}

ln(1/3) = -.2t

t = 5.49 hr

The minimum temperature that corresponds to this is:

T = 100(5.49) + 1500e^{-0.2(5.49)} = 1049^{o}

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