Activity 1.5.3.
A potato is placed in an oven whose temperature is 350 degrees Fahrenheit, and the potato’s temperature \(F\) (in degrees Fahrenheit) is recorded in the following table. Time \(t\) is measured in minutes.
1
Thanks to Nick Owad of Hood College for conducting experiments with actual potatoes in his oven in order to generate the data for this activity.
| \(t\) | \(0\) | \(10\) | \(20\) | \(30\) | \(40\) | \(50\) | \(60\) |
| \(F(t)\) | \(65\) | \(94.7\) | \(141.7\) | \(167.6\) | \(182.4\) | \(197.9\) | \(209.3\) |
(a)
Use a central difference to estimate the instantaneous rate of change of the potato’s temperature at \(t= 20\text{.}\) Include units on your answer.
(b)
Use a central difference to estimate the instantaneous rate of change of the potato’s temperature at \(t= 40\text{.}\) Include units on your answer.
(c)
Without doing any calculation, which do you expect to be greater: \(F'(50)\) or \(F'(60)\text{?}\) Why?
(d)
Suppose we know that \(F(46) = 192.5\) and \(F'(46) = 1.39\text{.}\) What are the respective units on these two quantities? What do you expect the temperature of the potato to be when \(t = 47\text{?}\) when \(t = 48\text{?}\) Why?
(e)
Write a couple of careful sentences that describe the behavior of the temperature of the potato on the time interval \([10,40]\text{,}\) as well as the behavior of the instantaneous rate of change of the potato’s temperature on the same time interval.
