Problem 65
The XYZ Company manufactures wicker chairs. With its present machines, it has a maximum yearly output of 500 units. If it makes \(x\) chairs, it can set a price of \(p(x)=200-0.15 x\) dollars each and will have a total yearly cost of \(C(x)=5000+6 x-0.002 x^{2}\) dollars. The company has the opportunity to buy a new machine for \(\$ 4000\) with which the company can make up to an additional 250 chairs per year. The cost function for values of \(x\) between 500 and 750 is thus \(C(x)=9000+6 x-0.002 x^{2} .\) Basing your analysis on the profit for the next year, answer the following questions. (a) Should the company purchase the additional machine? (b) What should be the level of production?
Problem 67
The ZEE Company makes zingos, which it markets at a price of \(p(x)=10-0.001 x\) dollars, where \(x\) is the number produced each month. Its total monthly cost is \(C(x)=200+4 x-0.01 x^{2} .\) At peak production, it can make 300 units. What is its maximum monthly profit and what level of production gives this profit?
Problem 69
The arithmetic mean of the numbers \(a\) and \(b\) is \((a+b) / 2\), and the geometric mean of two positive numbers \(a\) and \(b\) is \(\sqrt{a b} .\) Suppose that \(a>0\) and \(b>0\). (a) Show that \(\sqrt{a b} \leq(a+b) / 2\) holds by squaring both sides and simplifying. (b) Use calculus to show that \(\sqrt{a b} \leq(a+b) / 2 .\) Hint: Consider \(a\) to be fixed. Square both sides of the inequality and divide through by \(b .\) Define the function \(F(b)=(a+b)^{2} / 4 b\). Show that \(F\) has its minimum at \(a\). (c) The geometric mean of three positive numbers \(a, b\), and \(c\) is \((a b c)^{1 / 3} .\) Show that the analogous inequality holds: $$ (a b c)^{1 / 3} \leq \frac{a+b+c}{3} $$ Hint: Consider \(a\) and \(c\) to be fixed and define \(F(b)=\) \((a+b+c)^{3} / 27 b .\) Show that \(F\) has a minimum at \(b=\) \((a+c) / 2\) and that this minimum is \([(a+c) / 2]^{2}\). Then use the result from (b).
