Bears search for berries that grow in patches that can be spread out across a large...

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Bears search for berries that grow in patches that can be spread out across a large area. A bear will spend time in one patch gathering food before moving to another patch. The number of berries collected in a patch depends on the amount of time spent in the patch: B(t) = At" where B(t) is the number of berries collected at the end of t hours spent in the k+tn patch. The constants A and k are positive, and n is a positive constant. The values of A, n, and k vary for different bears and different patches. 1a. Suppose a particular patch has 1,000 berries, and it takes the bear one hour to collect 500 berries. Which of the constants (A, k, and n) can you determine from this information, and what are they? Explain how you got your answer. 1b. Suppose a particular bear likes to settle into a berry patch before it really starts eating. So, when it first reaches a patch (when t 0), its rate of berry finding is approximately zero. What does this tell you about the value of n? 1c. Using the definition of the derivative, calculate dB when n dt 1 for arbitrary A and k. (Your answer should depend on %3D the unspecified constants A and k, as well as the variable t.) 1d. For what values of t is the derivative you calculated positive? negative? What happens to this derivative as time goes to infinity? What do these tell you about the bear's foraging? 1e. Relate the difficulty of finding berries to the derivative of B(t). That is, what kinds of derivatives tell you it's tough to find berries, and what kinds of derivatives tell you it's easy? Bears search for berries that grow in patches that can be spread out across a large area. A bear will spend time in one patch gathering food before moving to another patch. The number of berries collected in a patch depends on the amount of time spent in the patch: B(t) = At" where B(t) is the number of berries collected at the end of t hours spent in the k+tn patch. The constants A and k are positive, and n is a positive constant. The values of A, n, and k vary for different bears and different patches. 1a. Suppose a particular patch has 1,000 berries, and it takes the bear one hour to collect 500 berries. Which of the constants (A, k, and n) can you determine from this information, and what are they? Explain how you got your answer. 1b. Suppose a particular bear likes to settle into a berry patch before it really starts eating. So, when it first reaches a patch (when t 0), its rate of berry finding is approximately zero. What does this tell you about the value of n? 1c. Using the definition of the derivative, calculate dB when n dt 1 for arbitrary A and k. (Your answer should depend on %3D the unspecified constants A and k, as well as the variable t.) 1d. For what values of t is the derivative you calculated positive? negative? What happens to this derivative as time goes to infinity? What do these tell you about the bear's foraging? 1e. Relate the difficulty of finding berries to the derivative of B(t). That is, what kinds of derivatives tell you it's tough to find berries, and what kinds of derivatives tell you it's easy?