91Ó°ÊÓ

Four feet of wire is to be used to form a square and a circle. (a) Express the sum of the areas of the square and the circle as a function \(A\) of the side of the square \(x\) (b) What is the domain of \(A ?\) (c) Use a graphing utility to graph \(A\) on its domain. (c) How much wire should be used for the square and how much for the circle in order to enclose the least total area? the greatest total area?

Identify the point of diminishing returns for the input- output function. For each function, \(R\) is the revenue and \(x\) is the amount spent on advertising. Use a graphing utility to verify your results. \(R=\frac{1}{50,000}\left(600 x^{2}-x^{3}\right), \quad 0 \leq x \leq 400\)

Biology: Wildlife Management The state game commission introduces 30 elk into a new state park. The population \(N\) of the herd is modeled by \(N=[10(3+4 t)] /(1+0.1 t)\) where \(t\) is the time in years. (a) Find the size of the herd after \(5,10,\) and 25 years. (b) According to this model, what is the limiting size of the herd as time progresses?

Average Profit The cost and revenue functions for a product are \(C=25.5 x+1000\) and \(R=75.5 x\) (a) Find the average profit function \(\bar{P}=\frac{R-C}{x}\) (b) Find the average profits when \(x\) is \(100,500,\) and 1000 . (c) What is the limit of the average profit function as \(x\) approaches infinity? Explain your reasoning.

Medicine The spread of a virus can be modeled by \(N=-t^{3}+12 t^{2}, \quad 0 \leq t \leq 12\) where \(N\) is the number of people infected (in hundreds), and \(t\) is the time (in weeks). (a) What is the maximum number of people projected to be infected? (b) When will the virus be spreading most rapidly? (c) Use a graphing utility to graph the model and to verify your results.