# Solving Optimization Problems - Page 6

Questions

- How do you find the volume of the largest right circular cone that can be inscribed in a sphere of radius r?
- How do you find the shape of a rectangle of maximum perimeter that can be inscribed in a circle of radius 5 cm?
- How do you minimize and maximize #f(x,y)=x^2y-xy# constrained to #3<x+y<5#?
- How do you maximize the volume of a right-circular cylinder that fits inside a sphere of radius 1 m?
- How do you minimize and maximize #f(x,y)=x/y-xy# constrained to #0<x-y<1#?
- A box has a bottom with one edge 7 times as long as the other. If the box has no top and the volume is fixed at V, what dimensions minimize the surface area?
- How do you minimize and maximize #f(x,y)=(xy)^2-x+y# constrained to #1<yx^2+xy^2<16#?
- What dimensions of the rectangle will result in a cylinder of maximum volume if you consider a rectangle of perimeter 12 inches in which it forms a cylinder by revolving this rectangle about one of its edges?
- How do you maximize and minimize #f(x,y)=x^2+9y^2-xy# constrained to #3<xy<5#?
- How do you maximize and minimize #f(x,y)=x-xy^2# constrained to #0<=x^2+y<=1#?
- How do you find the dimensions of the aquarium that minimize the cost of the materials if the base of an aquarium with volume v is made of slate and the sides are made of glass and the slate costs five times as much (per unit area) as glass?
- Find the dimensions that will minimize the cost of the material?
- A gutter is to be made from metal sheet where the length of the two of the two sides of the gutter is 4cm and the third side is 5cm. How do you find an angle so that the gutter will carry the maximum amount of water?
- A rectangular box is to be inscribed inside the ellipsoid #2x^2 +y^2+4z^2 = 12#. How do you find the largest possible volume for the box?
- A company can sell 5000 chocolate bars a month at $0.50 each. If they raise the price to $0.70, sales drop to 4000 bars per month. The company has fixed costs of $1000 per month and $0.25 for manufacturing each bar. What price will maximize the profit?
- How do solve applications of differentiation?
- How do you find the point on the graph of #x^2−2x=15−y^4# has the largest y-coordinate?
- A market survey suggests that, on the average, one additional unit will remain vacant for each 3 dollar increase in rent. Similarly, one additional unit will be occupied for each 3 dollar decrease in rent. What rent should the manager charge to maximize?
- What is the maximum volume of the box, given the parameters below?
- How do you minimize and maximize #f(x,y)=x+y# constrained to #0<x+3y<2#?