Expert: Paul Klarreich - 2/24/2008

Question
Suppose you put a potato in a hot oven, maintained at a constant temperature of 200 degrees Celsius.  As the potato picks up heat from the oven, it's temperature rises.

A) Draw a possible graph of the temperature T of the potato against time t (minutes) since it is put in the oven.  Explain any interesting features of the graph, and in particular explain it's concavity.

B) Suppose that, at t = 30, the temperature T of the potato is 120 degrees and increasing at the (instantaneous) rate of  2 degrees per minute.  Using this information plus what you know about the shaped of the T graph, estimate the temperature at t = 40.

C) Suppose in addition you are told that at t=60, the temperature of the potato is 165 degrees.  Can you improve your estimate of the temperature at t=40?

D) Assuming all the data given so far, estimate the time at which the temperature of the potato is 150 degrees.

My comments: I feel generally able to estimate the time when given a temperature.  However, step one throws me off.  What would a graph of a cooking potato look like?  I don't see why it would not be a straight line and this means I can't answer the subsequent problems until I've been able to graph this.

Answer
Questioner:   Madeline
Category:  Calculus
Private:  No
 
Subject:  Section: Short Cuts to Differentiation
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Hi, Maddy,

Question:  Suppose you put a potato in a hot oven, maintained at a constant temperature of 200 degrees Celsius.  As the potato picks up heat from the oven, it's temperature rises.

>> Smart.

A) Draw a possible graph of the temperature T of the potato against time t (minutes) since it is put in the oven.  Explain any interesting features of the graph, and in particular explain it's concavity.

>> "its", please -- no apostrophe here. [gotta watch those details.]

It will be hard to draw a graph, but, you can be sure that T(t) will be asymptotic to the horizontal line T = 200.  It will also be turning to the right (concave down).  That is because the rate of increase of T (a.k.a. dT/dt) gets smaller as T - 200 decreases.  The less the difference between 200 (oven) and T(t) (potato) the less heat transfers to the potato and the less the increase in T(t).  So dT/dt is decreasing, thus d2T/dt2 is negative, thus the graph is concave down.


B) Suppose that, at t = 30, the temperature T of the potato is 120 degrees and increasing at the (instantaneous) rate of  2 degrees per minute.  Using this information plus what you know about the shaped of the T graph, estimate the temperature at t = 40.

>> That is 10 seconds later, so we would ESTIMATE a 2(10) = 20 deg increase, to 140. BUT I think it will be a bit lower, because of the discussion in (A).

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C) Suppose in addition you are told that at t=60, the temperature of the potato is 165 degrees.  Can you improve your estimate of the temperature at t=40?


         dT   T2 - T1   165 - 120   45
Estimate: -- = ------- = --------- = -- = 1.5 deg/min.
         dt   t2 - t1    60 - 30    30

New estimate for t = 40 = 120 + 1.5(10) = 120 + 15 = 135.  [I told you it would be a bit lower.]

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D) Assuming all the data given so far, estimate the time at which the temperature of the potato is 150 degrees.

>> That is an increase of 30 degrees over t = 30.  That would take 30/1.5 = 20 seconds, so it should reach 150 at 30 + 20 = 50 seconds.

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My comments: I feel generally able to estimate the time when given a temperature.  

>> Good. Your time in Junior High cooking class was not wasted.  You can look forward to a well-fed and happy husband.
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However, step one throws me off.  What would a graph of a cooking potato look like?  I don't see why it would not be a straight line and this means I can't answer the subsequent problems until I've been able to graph this.

>> Well, I hope the discussion in A clarified that.  Basically, the rate of increase of T decreases as T increases.  Since T increases with t,  dT/dt, which is the slope, is decreasing.  Since the slope is changing, the graph IS NOT A STRAIGHT LINE, and that is what calculus is all about.