A hot brick is removed from a kiln and set on the floor tocool. Let t be time in

Question

A hot brick is removed from a kiln and set on the floor tocool. Let t be time in minutes after the brick wasremoved. The difference, D(t), between the brick'stemperature, initially 350o F, and room temperature,70o F, decays exponentially over time at a rate of 3%per minute. The bricks's tempuerature H(t), is atransformation of D(t). Find a formula forH(t). Compare the graphs of D(t) andH(t), paying attention to the asymptotes. A hot brick is removed from a kiln and set on the floor tocool. Let t be time in minutes after the brick wasremoved. The difference, D(t), between the brick'stemperature, initially 350o F, and room temperature,70o F, decays exponentially over time at a rate of 3%per minute. The bricks's tempuerature H(t), is atransformation of D(t). Find a formula forH(t). Compare the graphs of D(t) andH(t), paying attention to the asymptotes.

Explanation / Answer

I will take a stab at this problem since it has a lot ofviews. Hopefully if my thinking is wrong or I left out astep, this might jump start someone else on the correct path orhelp you on your way. From what I remember, the exponential decay curve is aninitial temperature T(0), multiplied by e-rt wherer is a constant in the system (the 3% here) and t is the timeinterval, minutes in this problem. So for D(t) I obtain: D(t) = D(0) * e-rt => D(0) is the initialdifference, 350deg. - 70deg. = 280deg. D(t) = 280deg. * e-(.03*t) H(t) will have the same format as D(t), but will be verticallyshifted 70 degrees: H(t) = D(t) + 70deg. = (280deg. * e-(.03*t)) +70deg. If you plot both functions on the same axis with D(t)/H(t) asthe y-axis and t as the x-axis you get two decay curves withhorizontal asymptotes about the x-axis. The formulas both work in that as t approaches infinity, thedifference equation, D(t), approaches 0deg. (brick androom are equal) and the brick temperature, H(t),approaches 70deg, the room temperature. These are theasymptotes of the equations. Does this make sense? H(t) will have the same format as D(t), but will be verticallyshifted 70 degrees: H(t) = D(t) + 70deg. = (280deg. * e-(.03*t)) +70deg. If you plot both functions on the same axis with D(t)/H(t) asthe y-axis and t as the x-axis you get two decay curves withhorizontal asymptotes about the x-axis. The formulas both work in that as t approaches infinity, thedifference equation, D(t), approaches 0deg. (brick androom are equal) and the brick temperature, H(t),approaches 70deg, the room temperature. These are theasymptotes of the equations. Does this make sense?

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