# Solving Optimization Problems - Page 3

Questions

- A piece of wire 60 cm in length is cut into two pieces. The first piece forms a rectangle 5 times as wide as it is long. The second piece forms a square. Where should the wire be cut to? 1)minimize the total area 2)maximize the total area
- The point #P# lies on the #y#-axis and the point #Q# lies on the #y#-axis. A triangle is formed by connecting the origin #O# to #P# and #Q#, If #PQ=23# then prove that the maximum area occurs when when #OP=OQ#?
- How do you minimize and maximize #f(x,y)=2x^2-x/(2x-3y)# constrained to #1<yx^2+xy^2<16#?
- An open top box is to be constructed so that its base is twice as long as it is wide. Its volume is to be 2400cm cubed. How do you find the dimensions that will minimize the amount of cardboard required?
- A rancher has 1000m of fencing to enclose two rectangular corrals. The corrals have the same dimensions and one side in common (let that side be x). What dimensions will maximize the enclosed area?
- How do you minimize and maximize #f(x,y)=(x-y)/((x-2)^2(y-4))# constrained to #xy=3#?
- How do you minimize and maximize #f(x,y)=x^3-y^2-xy# constrained to #xy=4#?
- A fence 8 ft tall runs parallel to a tall building at a distance of 4 ft from the building. What is the length of the shortest ladder that will reach from the ground over the fence to the wall of the building?
- How do you maximize and minimize #f(x,y)=1/x+y^2+1/(xy)# constrained to #2<x/y<4#?
- How do you find the value of x that gives the minimum average cost, if the cost of producing x units of a certain product is given by #C = 10,000 + 5x + (1/9)x^2#?
- What are the radius, length and volume of the largest cylindrical package that may be sent using a parcel delivery service that will deliver a package only if the length plus the girth (distance around) does not exceed 108 inches?
- How do you minimize and maximize #f(x,y)=x/e^(x-y)+y# constrained to #1<x^2/y+y^2/x<9#?
- How do you maximize and minimize #f(x,y)=xy-y^2# constrained to #0<x+3y<4#?
- A piece of wire 20 m long is cut into two pieces. One piece is bent into a square and the other is bent into an equilateral triangle. How much wire should be used for the square in order to maximize the total area?
- What is the smallest perimeter possible for a rectangle of area 16 in^2?
- How do you find the area of the largest rectangle that can be inscribed in a right traingle with legs of lengths 3 cm and 4cm if two sides of the rectangle lie along the legs?
- How do you optimize #f(x,y)=xy-x^2+e^y# subject to #x-y=8#?
- How do you minimize and maximize #f(x,y)=(x-2)^2/9+(y-3)^2/36# constrained to #0<x-y^2<5#?
- How do you find two positive numbers whose product is 192 and the sum is a maximum?
- Find the dimension of the box that will minimize the total cost. Find the minimum total cost ?