The temperature T, in degrees Farenheight, of a yam after sitting in a hot oven
ID: 2855590 • Letter: T
Question
The temperature T, in degrees Farenheight, of a yam after sitting in a hot oven for t minutes is given by the function T(t)= 350 - 280e-0.2t . (A) What is the intital temperature of the yam? (B) Given that over time the temperature of the yam will approach the temperature inside the oven, use a limit to determine the temperature of the oven. (C) How long will it take for the yam to be within 5 degrees Farenheight of the temperature of the oven? (D) The first derivative of T(t) measures the rate of change of the temperature of the yam. The second derivative of T(t) measures the rate of change of the rate of change of the temperature of the yam. Use T'(t) and T''(t) to argue thta the temperature of the yam increases at a decreasing rate. This statement is related to the saying "Cold water boils faster." How?
Explanation / Answer
T(t)= 350 - 280e-0.2t
A) Initial temperature ==> at t = 0
T(0) = 350 - 280e-0.2(0) = 350 - 280 = 70 Fahrenheit
B) lin [t -> ] 350 - 280e-0.2t = 350 - 280(0) = 350 Fahrenheit
C) T(t) = 345
==> 345 = 350 - 280e-0.2t
==> 280e-0.2t = 5
==> e-0.2t = 5/280
==> e-0.2t = 1/56
apply natural log on both sides
ln e-0.2t = ln (1/56)
==> -0.2t = -4.0254
==> t = -4.0254/-0.2 = 20.13 minutes
D) T(t) = 350 - 280e-0.2t
T '(t) = 0 - 280e-0.2t(-0.2)
==> T '(t) = 56e-0.2t
==> T ''(t) = 56e-0.2t(-0.2)
==> T ''(t) = -11.2e-0.2t
T '(t) > 0 and T ''(t) < 0 .Hence the temperature of the yam increases at a decreasing rate.
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