1.A company is informed by its market analyst, that if the price of widgets is $90-x, the company will sell 100+10x thousand widgets. At what price should the widget be set so the company maximizes income?
2.Another company sells wrought iron umbrella trees. This company has fixed costs of $100,000/month. The cost in labor and material is $15 per umbrella sold per month. To sell x items, the price must be set at $50-0.7x, where x is in thousands of umbrellas sold per month. What price will maximize profit?
3.A drum in the form of a circular cylinder and open at one of the circular ends, is to be made so as to contain one cubic yard. Find the dimensions of the drum (height h and base radius r ) which minimizes the amount of material going into the drum. The surface area of the drum includes the area of the cylindrical side and the circular disc at bottom.
4.A rectangle with sides parallel to to the coordinate axes is inscribed in the ellipse
16x^2 + 9y^2 = 144
Find the dimensions of the rectangle of greatest area.