SECTION 45 ExponentilEqutions Getting Iinformation from a Modet (a) Find the yea

Question

SECTION 45 ExponentilEqutions Getting Iinformation from a Modet (a) Find the yearly owith fastor a (b) Find an exponential goth t )Ca for the population f years (c) How long will it take for the populstion to dosble? (d) Use the model found in paut dhy to predict the year in which the population since 2003 will reach 90 million. 56. Population of Germany In 2004 the population of Glermany was about with an annual growth rate of 002% . Assume that this rate of growth (a) Find the yearly growth factor a (b) Find an exponential growth model) Ca for the popu (c) How long will it take for the population to double? years since 2004. ) to predict the year in which the population will 57. Pot of Chill Angela prepares a large pot of chili the night before a before it can d) Use the model found in part (b reach 83 million. church potluck. be ht before The temperature of the chili is 212°F, and it must cool down to 70 be stored in the refrigerator. Assume that the ambient temperature is 65°F and the heat transfer coefficient is k = 2.895. (a) Find a model for the temperature T of the pot of chili hours after cooling. (b) How (b) How long will it take for the pot of chiüli to cool down to the desired temperatu 70°F? d (b). (c) Graph the function T to confirm your answers to parts (a) an 58. Time of Death Newton's Law of Cooling is used in homicide investigations to location whose determine the time of death. Suppose that a body is discovered in a ambient temperature is 60°F. The police determine that the heat transfer coefficient in this case is k -0.1947. (The heat transfer coefficient depends on many factors, inclding the size of the body and the amount of clothing.) Normal body temperature is 98.6°F (a) Find a model for the temperature T of the body t hours after death. (b) When the body was found, t had a temperature of 72°F. Find the length of time t victim has been dead. (c) Graph the function T to confirm your answers to parts (a ) and (b). Boiling Water A kettle of water is brought to a boil in a room where the temperature s 20°C, After 15 minutes the temperature of the water has decreased from 100°C to 5 ) Find the heat transfer coefficient k, and find a model for the temperature T of the water t hours after it is brought to a boil. Use the model to predict the temperature of the water after 25 minutes. Illustrate by graphing the temperature function. How long will it take the water to e-

Explanation / Answer

The temperature surplus of the body with respect to the room decresases at an exponential rate; And will never fall below the room temperature;
Thus, T(t) = (T0 - TR) e-kt + TR (here the term "+TR" is added to show that TR is the minimum temperature that the body can fall to;
(Note: here 't' is in time units)
b) T(t) = 72
TR =60
T0 = 98.6 and
k = 0.1947
Substituting in the equation we get:
72 = (98.6 - 60) e-(0.1947)(t) + 60
38.6 e-0.1947t = 12
e0.1947t = 38.6/12 = 3.2166
0.1947t = ln (3.2166)
0.1947t =1.16834
t= 1.16834/ 0.1947 = 6

Thus, the victim has been dead for a time period of 6 units

Note: since the units of the value 0.1947 have not be specified , we cannot specific the units of 't'
The units of 't' are the inverse of the units of the k value;

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