4. (a] Solve the following equation. Clearly explain whether all solutions are acceptable. 72x + 6x-5 = 343 [b) Solve the following equation. Clearly explain whether all solutions are acceptable. + 2 (c) Solve the following equation. Clearly explain whether all solutions are acceptable. [Hint: Use properties of logarithms to condense the expression and write it as a single logarithm of a function with coefficient 1. What is the domain of the function?] logz (x* + 2x2 - x - 2) - log, (x + 1) - 2 log=(x3 + x - 2) = -71082 x + 15. (2) A species is in danger of extinction; its population is decreasing exponentially as P = Piper, where P. Po, & are, respectively, the current population, the original population (at timer = 0], and the rate of decay in appropriate units. (1 If 10 years ago we had P = 1500, and today we have P = 1000, find the decay rate. What does its sign signify? (in) if the population of the species drops below 100, the situation wil be incversible. When will this happen? (b] The radioactive element potassium 40 decays exponentially with a half-life of 1.31 billion years. (0 Find its decay rate. (i) The age of a dinosaur discovered in 1964 was estimated using potassium-40 dating of rocks surrounding the bones. The analysis indicated that 945% of the original amount of potassium-40 was still present. Estimate the age of the bones of the dinosaur. (c) Exponential growth is possible only when the available natural resources cannot be exhausted, which is not true in the real world. Charles Darwin recognized this fact; in the "struggle for existence", he stated that individuals will compete [with members of their own or other species] for limited resources. The successful ones will survive to pass on their own characteristics and traits to the next generation at a greater rate, a process known as natural selection. To model the reality of limited resources, population ecologists developed the logistic growth model In the model, while the original growth has an exponential behavior, this growth will level off as resources become scarce. The population sing at this plateau represents the maximum population size that a particular environment can support; it is called the carrying capacity (KJ. In other words, the carrying capacity of an organism in a given environment is defined to be the maximum population of the organism that the environment can sustain indefinitely. 7The expression we use to calculate logistic growth adds the carrying capacity as a moderating force in the growth rate. If the population as a function of time is P(t), the expression K - P indicates how many individuals may be added to a population at a given stage, and -is the K-P fraction of the carrying capacity available for further growth. Thus, the exponential growth model (in which the change with of the population is proportional to the rate of change and the value of the population itself) is restricted by this factor to generate the logistic growth equation: Kert P(t) = Po 7 K - Po) + Poert. (i) The current population (P,) of the Earth is 7.8 billion people. The current rate of growth (r) is 1.05%. The carrying capacity (K ) varies, depending on many factors; let's assume it is 40 billion people. (Note: People in different parts of the world are consuming different amounts of the Earth's resources. If everyone on Earth lived like a middle-class American, consuming roughly 3.3 times the subsistence level of food and about 250 times the subsistence level of clean water, the Earth could only support about 2 billion people. On the other hand, if everyone on the planet consumed only what he or she needed, 40 billion would be a feasible number.) When would the Earth's population be double the current value? (ii) When would the Earth's population reach 90%% of its carrying capacity? (iii) Plot the function P(t). (iv) Let us assume that we do not act until after we go beyond the carrying capacity. Let us start with a current population that is 25% above carrying capacity (that is, P. = 50 billion people), with everything else remaining the same. What does the logistic growth equation predict? Plot the function P(() and find how many years it will take to get within 5% of the carrying capacity