1. Estimating the Temperature of a Yam
Suppose you put a yam in a hot oven, maintained at a constant temperature of 200�C. As
the yam picks up heat from the oven, its temperature rises.16
(a) Draw a possible graph of the temperature T of the yam against time t (minutes) since it is
put into the oven. Explain any interesting features of the graph, and in particular explain its
concavity.
(b) Suppose that, at t = 30, the temperature T of the yam is 120� and increasing at the (instantaneous)
rate of 2�/min. Using this information, plus what you know about the shape of the
T graph, estimate the temperature at time t = 40.
(c) Suppose in addition you are told that at t = 60, the temperature of the yam is 165�. 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 yam
To estimate the temperature of the yam at different times and analyze its behavior, we can use calculus and the concepts of rates of change and concavity.
a) Graphing the Temperature vs. Time:
To draw a possible graph, we need to consider the information given. The temperature of the yam initially is lower than the oven temperature. As time passes, the yam absorbs heat from the oven and its temperature rises. The graph will have an increasing trend as time goes on.
Since the yam is picking up heat from the oven, we know that the temperature will approach the oven temperature of 200ºC but will never surpass it. Therefore, we can draw an asymptote at T = 200ºC.
The concavity of the graph depends on the rate at which the yam's temperature increases. If the rate of temperature increase is constant, the graph will be a straight line. If the rate of temperature increase is itself increasing, the graph will be concave upward (like a U-shape). If the rate of temperature increase is decreasing, the graph will be concave downward (like an inverted U-shape).
b) Estimating the temperature at t = 40:
We are given that at t = 30, the temperature of the yam is 120ºC and increasing at the rate of 2ºC/min. Since we know the shape of the graph is increasing, we can assume it is concave upward.
Now, to estimate the temperature at t = 40, we can use the given information along with the assumption that the concavity remains the same. If the rate of increase is 2ºC per minute, we can estimate an additional increase of 2ºC over 10 minutes.
Therefore, at t = 40, the estimated temperature would be 120ºC + (2ºC/minute) * 10 minutes = 140ºC.
c) Improving the estimate at t = 40:
Given that at t = 60, the temperature is 165ºC, we can now improve our estimate at t = 40. Since we know the temperature is increasing, and at a higher rate, between t = 40 and t = 60, we can assume that the temperature at t = 40 is less than 140ºC. Therefore, our previous estimate of 140ºC is an overestimate.
d) Estimating the time at which the yam reaches 180ºC:
Although the question does not explicitly ask for this, we can estimate the time at which the temperature of the yam reaches 180ºC based on the given information.
Since we know the yam's temperature is increasing at a rate of 2ºC/min at t = 30 and the graph is concave upward, we can assume the rate of increase is constant between t = 30 and t = 60. Therefore, it took 30 minutes for the temperature to increase from 120ºC to 165ºC, which means it took an additional 15 minutes to increase by 45ºC.
Using this constant rate of increase, we can estimate that it will take another 15 minutes to increase from 165ºC to 180ºC. Therefore, the estimated time at which the temperature of the yam reaches 180ºC is t = 75 minutes.