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
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a) To draw a possible graph of the temperature T of the yam against time t, we know that the yam is put into a hot oven maintained at a constant temperature of 200°C. As the yam picks up heat from the oven, its temperature rises. Therefore, initially, the yam's temperature will be lower than 200°C and will gradually increase over time until it reaches the same temperature as the oven.
The graph will start at a lower initial temperature and will then steadily increase as time goes on. It should eventually level off when it reaches the temperature of the oven. The graph will be concave upwards since the rate of increase of temperature will slow down as the yam approaches the oven temperature.
b) At t = 30, the temperature T of the yam is 120°C and increasing at a rate of 2°C/min. We can use this information to estimate the temperature at t = 40. Since the rate of increase is constant, we can assume that in 10 minutes (from t = 30 to t = 40), the temperature will increase by 2°C. Therefore, we can estimate the temperature at t = 40 to be 120°C + 2°C = 122°C.
c) With the additional information that at t = 60, the temperature of the yam is 165°C, we can improve our estimate of the temperature at t = 40. We know that the rate of increase is constant, so if the temperature increases by 2°C per minute, then in 20 minutes (from t = 40 to t = 60), the temperature will increase by 40°C (2°C/min x 20 min). Therefore, we can estimate the temperature at t = 40 to be 122°C + 40°C = 162°C.
d) Assuming all the data given so far, we can estimate the time at which the temperature of the yam reaches the oven temperature of 200°C. Since the rate of increase is constant at 2°C/min, we can calculate the time it takes to increase the temperature by 80°C (200°C - 120°C). Therefore, it will take 80°C / 2°C/min = 40 minutes for the temperature of the yam to reach 200°C. So, the estimated time at which the temperature of the yam reaches 200°C is t = 40 minutes.