Solving Optimization Problems

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Questions

• How do you find two numbers whose difference is 100 and whose product is a maximum?
• How do you find the dimensions of a rectangle whose area is 100 square meters and whose perimeter is a minimum?
• How do you find the points on the ellipse #4x^2+y^2=4# that are farthest from the point #(1,0)#?
• How do you find the dimensions of the rectangle with largest area that can be inscribed in a semicircle of radius #r# ?
• Question #517b1
• Let's say I have $480 to fence in a rectangular garden. The fencing for the north and south sides of the garden costs $10 per foot and the fencing for the east and west sides costs $15 per foot. How can I find the dimensions of the largest possible garden.?
• 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 dimensions of a rectangular box that has the largest volume and surface area of 56 square units?
• What are the dimensions of a box that will use the minimum amount of materials, if the firm needs a closed box in which the bottom is in the shape of a rectangle, where the length being twice as long as the width and the box must hold 9000 cubic inches of material?
• How do you find the dimensions that minimize the amount of cardboard used if a cardboard box without a lid is to have a volume of #8,788 (cm)^3#?
• How do you find the dimensions of the box that minimize the total cost of materials used if a rectangular milk carton box of width w, length l, and height h holds 534 cubic cm of milk and the sides of the box cost 4 cents per square cm and the top and bottom cost 8 cents per square cm?
• How do you find two positive numbers whose sum is 300 and whose product is a maximum?
• How do you find two positive numbers whose product is 750 and for which the sum of one and 10 times the other is a minimum?
• How do you construct a 12-ounce cylindrical can with the least amount of material?
• 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?
• What are the coordinates of the points a and b and minimize the length of the hypotenuse of a right triangle that is formed in the first quadrant by the x-axis, the y-axis, and a line through the point (1,2) where point a is at (0,y) and point b is at (x,0)?
• What will the dimensions of the resulting cardboard box be if the company wants to maximize the volume and they start with a flat piece of square cardboard 20 feet per side, and then cut smaller squares out of each corner and fold up the sides to create the box?
• What is the maximum possible area of the rectangle that is to be inscribed in a semicircle of radius 8?
• What dimensions will result in a box with the largest possible volume if an open rectangular box with square base is to be made from #48 ft^2# of material?
• How do you find two nonnegative numbers whose sum is 9 and so that the product of one number and the square of the other number is a maximum?
• What height h and base radius r will maximize the volume of the cylinder if the container in the shape of a right circular cylinder with no top has surface area #3pi ft^2#?
• 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 find the dimensions (radius r and height h) of the cone of maximum volume which can be inscribed in a sphere of radius 2?
• How do you find the point on the graph of #x^2−2x=15−y^4# has the largest y-coordinate?
• What is the smallest possible value of the sum of their squares if the sum of two positive numbers is 16?
• What is the largest area that can be fenced off of a rectangular garden if it will be fenced off with 220 feet of available material?
• How do you find the cost of materials for the cheapest such container given a rectangular storage container with an open top is to have a volume of #10m^3# and the length of its base is twice the width, and the base costs $10 per square meter and material for the sides costs $6 per square meter?
• How do you find two positive numbers whose sum is 300 and whose product is a maximum?
• 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 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 dimensions would you need to make a glass cage with maximum volume if you have a piece of glass that is 14" by 72"?
• How do you maximize the volume of a right-circular cylinder that fits inside a sphere of radius 1 m?
• How do you find the dimensions of the largest possible garden if you are given four hundred eighty dollars to fence a rectangular garden and the fencing for the north and south sides of the garden costs $10 per foot and the fencing for the east and west sides costs $15 per foot?
• How do you find the shape of a rectangle of maximum perimeter that can be inscribed in a circle of radius 5 cm?
• 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?
• How do you maximize a window that consists of an open rectangle topped by a semicircle and is to have a perimeter of 288 inches?
• How do you find two positive numbers whose product is 192 and the sum is a maximum?
• How do you find the point on the curve #y=2x^2# closest to (2,1)?
• The top and bottom margins of a poster are 4 cm and the side margins are each 6 cm. If the area of printed material on the poster is fixed at 384 square centimeters, how do you find the dimensions of the poster with the smallest area?
• A car rental agency rents 220 cars per day at a rate of 27 dollars per day. For each 1 dollar increase in the daily rate, 5 fewer cars are rented. At what rate should the cars be rented to produce the maximum income, and what is the maximum income?
• How do you find the points on the parabola #y = 6 - x^2# that are closest to the point (0,3)?
• A rectangle is constructed with it's base on the x-axis and the two of its vertices on the parabola #y=49 - x^2#. What are the dimensions of the rectangle with the maximum area?
• At noon, aircraft carrier Alpha is 100 kilometers due east of destroyer Beta. Alpha is sailing due west at 12 kilometers per hour. Beta is sailing south at 10 kilometers per hour. In how many hours will the distance between the ships be at a minimum?
• A cylinder is inscribed in a right circular cone of height 6 and radius (at the base) equal to 5. What are the dimensions of such a cylinder which has maximum volume?
• An open -top box is to be made by cutting small congruent squares from the corners of a 12-by12-in. sheet of tin and bending up the sides. How large should the squares cut from the corners be to make the box hold as much as possible?
• What are the dimensions of the lightest open-top right circular cylindrical can that will hold a volume of 125 #cm^3#?
• 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 find the length and width of a rectangle whose area is 900 square meters and whose perimeter is a minimum?
• How do you find a positive number such that the sum of the number and its reciprocal is as small as possible?
• How do you find the area of the largest isosceles triangle having a perimeter of 18 meters?
• A farmer wants to fence an area of 6 million square feet in a rectangular field and then divide it in half with a fence parallel to one of the sides of the rectangle. How can he do this so as to minimize the cost of the fence?
• How do you find the point on the line #y=4x + 7# that is closest to the point (0,-3)?
• 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 maximize the perimeter of a rectangle inside a circle with equation: #x^2+y^2=1#?
• Which point on the parabola #y=x^2# is nearest to (1,0)?
• A rectangle is inscribed with its base on the x-axis and its upper corners on the parabola #y=6-x^2#. What are the dimensions of such a rectangle with the greatest possible area? thanks for any help!?
• A farmer owns 1000 meters of fence, and wants to enclose the largest possible rectangular area. The region to be fenced has a straight canal on one side, and thus needs to be fenced on only three sides. What is the largest area she can enclose?
• How do you find the points on the parabola #2x = y^2# that are closest to the point (3,0)?
• A physical fitness room consists of a rectangular region with a semicircle on each end. If the perimeter of the room is to be a 200 meter running track, how do find the dimensions that will make the area of the rectangular regions as large as possible?
• A cylinder has a volume of 300 cubic inches. The top and bottom parts of the cylinder cost $2 per square inch. And the sides of the cylinder cost $6 per square inch. What are the dimensions of the Cylinder that minimize cost based on these constraints?
• How do you find the point on the the graph #y=sqrtx# which is plosest to the point (4,0)?
• A box with a square base and open top must have a volume of 32,000cm^3. How do you find the dimensions of the box that minimize the amount of material used?
• A sector of a circle whose radius is r and whose angle is theta has a fixed perimeter P. How do you find the values of r and theta so that the area of the sector is a maximum?
• A rectangle with sides parallel to the axes is inscribed in the region bounded by the axes and the line x+2y = 2. How do you find the maximum area of this triangle?
• 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 rectangle is to have an area of 16 square inches. How do you find its dimensions so that the distance from one corner to the midpoint of a nonadjacent side is a minimum?
• A right cylinder is inscribed in a sphere of radius r. How do you find the largest possible volume of such a cylinder?
• 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 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?
• 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?
• A triangle has two equal sides 4 inches long. what is the length of the third side (the base) if the triangle is to have maximum area?
• If 2400 square centimeters of material is available to make a box with a square base and an open top, how do you find the largest possible volume of the box?
• A piece of wire 10 m long is cut into two pieces. One piece is bent into a square and the other is bent into an equilateral triangle. How should the wire be cut so that the total area enclosed is maximum?
• A piece of wire 10 m long is cut into two pieces. One piece is bent into a square and the other is bent into an equilateral triangle. How should the wire be cut so that the total area enclosed is minimum?
• How do you find the largest possible area for a rectangle inscribed in a circle of radius 4?
• A long rectangular sheet of metal, 12cm wide, is to be made into a rain gutter by turning up two sides which make an angle of 120 degrees with the base. How many cm should be turned up to give the gutter its greatest capacity?
• 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?
• A sodium chloride crystal in the shape of a cube is expanding at the rate of 60 cubic microns per second. How fast is the side of the cube growing when the volume is 1000 cubic microns?
• 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?
• How do you find the volume of the largest open box that can be made from a piece of cardboard 35 inches by 19 inches by cutting equal squares from the corners and turning up the sides?
• Among all right circular cones with a slant height of 12, what are the dimensions (radius and height) that maximize the volume of the cone?
• What is the smallest perimeter possible for a rectangle of area 16 in^2?
• What dimensions should you make a rectangular poster of total area 72 square inches in order to maximize the printed area if it is required that the top and bottom margins be 2 inches and the side be 1 inch?
• A rectangular sheet with perimeter of 33cm and dimensions x by y are rolled into a cylinder, x being the perimeter What values x and y give the largest volume?
• You are designing a rectangular poster to contain 50 in^2 of printing with a 4-in. margin at the top and bottom and a 2-in margin at each side. What overall dimensions will minimize the amount of paper used?
• A cylindrical can is to be made to hold 1000cm^3 of oil. How do you find the dimensions that will minimize the cost of metal to manufacture the can?
• A rectangular poster is to contain 108cm^2 of printed matter with margins of 6cm at the top and bottom and 2 cm on the sides. what's the least cost to make the poster if the printed material costs 5 cents/cm^2 and the margins are 1 cent/cm^2?
• A fence 4 feet tall runs parallel to a tall building at a distance of 2 feet 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 optimize #f(x,y)=2x^2+3y^2-4x-5# subject to #x^2+y^2=81#?
• How do you optimize #f(x,y)=x^2-3y^2# subject to #x^3-y^2=8#?
• How do you optimize #f(x,y)=xy-x^2+e^y# subject to #x-y=8#?
• How do you maximize and minimize #f(x,y)=e^-x+e^(-3y)-xy# subject to #x+2y<7#?
• How do you maximize and minimize #f(x,y)=1/x+y^2+1/(xy)# constrained to #2<x/y<4#?
• How do you maximize and minimize #f(x,y)=1/(xy)# constrained to #2<x^2+y<4#?
• How do you maximize and minimize #f(x,y)=xy-y^2# constrained to #0<x+3y<4#?
• How do you maximize and minimize #f(x,y)=x^2+3xy+9y^2# constrained to #0<x+3y<2#?
• 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)=5x+6y# constrained to #x>0, y>0, x+2y<8, x+y<5#?
• How do you minimize and maximize #f(x,y)=x^2y-xy# constrained to #3<x+y<5#?
• How do you minimize and maximize #f(x,y)=(x-y)/(x^3-y^3# constrained to #2<xy<4#?
• How do you minimize and maximize #f(x,y)=x^3-y# constrained to #x-y=4#?
• How do you minimize and maximize #f(x,y)=x^3-y^2-xy# constrained to #xy=4#?
• How do you minimize and maximize #f(x,y)=ye^x-xe^y# constrained to #xy=4#?
• How do you minimize and maximize #f(x,y)=(x-y)/x^2# constrained to #xy=4#?
• How do you minimize and maximize #f(x,y)=(xy)/((x-2)(y-4))# constrained to #xy=2#?
• 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)=xy^2e^y-e^x# constrained to #xy=1#?
• How do you minimize and maximize #f(x,y)=e^x/e^y-x^2y# constrained to #0<xy+y^2<1#?
• How do you minimize and maximize #f(x,y)=ye^(2x)-ln(y/x)# constrained to #0<xy-y+x<1#?
• How do you minimize and maximize #f(x,y)=sinx*cosy-tanx# constrained to #0<x-y<1#?
• How do you minimize and maximize #f(x,y)=(x^2+4y)/e^(y)# constrained to #0<x-y<1#?
• How do you minimize and maximize #f(x,y)=xe^x-y# constrained to #0<x-y<1#?
• How do you minimize and maximize #f(x,y)=x/y-xy# constrained to #0<x-y<1#?
• 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 minimize and maximize #f(x,y)=(x-2)^2-(y-3)^2/x# constrained to #0<xy-y^2<5#?
• How do you minimize and maximize #f(x,y)=x-y^2/(x+y)# constrained to #0<xy-y^2<5#?
• How do you minimize and maximize #f(x,y)=(x-y)/(x^2+y^2)# constrained to #0<xy-y^2<5#?
• How do you minimize and maximize #f(x,y)=(x-y)(x+y)+sqrt(xy)# constrained to #0<xy-y^2<5#?
• How do you minimize and maximize #f(x,y)=xsqrt(xy+y)# constrained to #0<xy-y^2<5#?
• How do you minimize and maximize #f(x,y)=x+y# constrained to #0<x+3y<2#?
• How do you minimize and maximize #f(x,y)=x^2-y/x# constrained to #0<x+3y<2#?
• How do you minimize and maximize #f(x,y)=x^2+y^(3/2)# constrained to #0<x+3y<2#?
• How do you minimize and maximize #f(x,y)=x^2+y^3# constrained to #0<x+3y<2#?
• How do you minimize and maximize #f(x,y)=x^2+y^3# constrained to #0<x+3xy<4#?
• How do you minimize and maximize #f(x,y)=x^2/y^3+xy^2# constrained to #0<x+3xy<4#?
• How do you minimize and maximize #f(x,y)=(x-y)^2e^x+x^2e^y/(x-y)# constrained to #1<yx^2+xy^2<3#?
• How do you minimize and maximize #f(x,y)=(2x-y)+(x-2y)/x^2# constrained to #1<yx^2+xy^2<3#?
• How do you minimize and maximize #f(x,y)=(2x-3y)^2-1/x^2# constrained to #1<yx^2+xy^2<3#?
• How do you minimize and maximize #f(x,y)=x^2+y^2# constrained to #1<yx^2+xy^2<16#?
• How do you minimize and maximize #f(x,y)=x^2/y+y^2/x# constrained to #1<yx^2+xy^2<16#?
• How do you minimize and maximize #f(x,y)=(xy)^2-x+y# constrained to #1<yx^2+xy^2<16#?
• How do you minimize and maximize #f(x,y)=sin^2y/x+cos^2x/y# constrained to #1<yx^2+xy^2<16#?
• How do you minimize and maximize #f(x,y)=x-y/(x-y/(x-y))# constrained to #1<yx^2+xy^2<16#?
• How do you minimize and maximize #f(x,y)=2x^2-x/(2x-3y)# constrained to #1<yx^2+xy^2<16#?
• How do you minimize and maximize #f(x,y)=(e^(yx)-e^(-yx))/(2yx)# constrained to #1<x^2/y+y^2/x<3#?
• 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)=x-siny# constrained to #0<=x+y<=1#?
• How do you maximize and minimize #f(x,y)=x^2-y/x# constrained to #0<=x+y<=1#?
• How do you maximize and minimize #f(x,y)=x-xy^2# constrained to #0<=x^2+y<=1#?
• How do you maximize and minimize #f(x,y)=(xy)^2-e^y+x# constrained to #0<=x-y<=1#?
• Question #b9993
• How do you find the dimensions of a rectangle with area 1000 m whose perimeter is as small as possible?
• How do you find the dimensions of the rectangle of largest area that can be inscribed in an equilateral triangle of side L if one side of the rectangle lies on the base of the triangle?
• An open top box is to have a rectangular base for which the length is 5 times the width and a volume of 10 cubic feet. It's five sides are to have as small a total surface area as possible. What are the sides?
• A square and a equilateral triangle are to be formed out of the same piece of wire. The wire is 6 inches long. How do you maximize the total area the square and the triangle contain?
• A right circular cylinder is inscribed in a cone with height 6m and radius 3m. How do you find the largest possible volume of such a cylinder?
• An observatory is to be in the form of a right circular cylinder surmounted by a hemispherical dome. If the hemispherical dome costs 5 times as much per square foot as the cylindrical wall, what are the most economic dimensions for a volume of 4000 cubic?
• How do I find the point on the graph f(x)=sqrt(x) closest to the point (4,0)? Please show the work
• Question #f78cb
• A farmer wants to enclose a 450,000m^2 rectangular field with fence. She then wants to sub-divide this field into three smaller fields by placing additional lengths of fence parallel to one of the side. How can she do this so that she minimizes the cost?
• A farmer has 160 feet of fencing to enclose 2 adjacent rectangular pig pens. What dimensions should be used so that the enclosed area will be a maximum?
• 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?
• A metal cylindrical can is to have a volume of 3.456pi cubic feet. How do you find the radius and height of the can which uses the smallest amount of metal?
• Question #eb797
• A fold is formed on a #20 cm × 30 cm# rectangular sheet of paper running from the short side to the long side by placing a corner over the long side. Find the minimum possible length of the fold?
• Find the maximum possible total surface area of a cylinder inscribed in a hemisphere of radius 1?
• An open-top box with a square base has a surface area of 1200 square inches. How do you find the largest possible volume of the box?
• Question #341dd
• Question #828cc
• Question #fa2ee
• What is the graphic for #f(x,y)=x y^(x/y)=C_0# ?
• Stevie completes a quest by travelling from #A# to #C# vi #P#. The speed along #AP# is 4 km/hour, and along #AB# it is 5 km/hour. Solve the following?
• Question #1be57
• Question #85982
• Find the dimensions that will minimize the cost of the material?
• Demand for rooms, of a hotel which has #58# rooms, is a function of price charged given by #u(p)=p^2-12p+45#. Find out at what price the revenue is maximized and what is the revenue?
• Question #05570
• Question #30b71
• Question #3498f
• Build a rectangular pen with three parallel partitions (meaning 4 total sections) using 500 feet of fencing. What dimensions will maximize the total area of the pen?
• What is the maximum volume of the box, given the parameters below?
• Functions 11 Word Problem?
• What is the maximum area of a rectangle that can be circumscribed about a given rectangle with length L and width W?
• Question #2041c
• Question #40c26
• Can anybody help me with this optimization problem?
• Question #e7bb3
• The area of a rectangle is #A^2#. Show that the perimeter is a minimum when it is square?
• A steel girder is taken to a 15ft wide corridor. At the end of the corridor there is a 90° turn, to a 9ft wide corridor. How long is the longest girder than can be turned in this corner?
• The productivity of a company during the day is given by # Q(t) = -t^3 + 9t^2 +12t # at time t minutes after 8 o'clock in the morning. At what time is the company most productive?
• #220# cars are rented at #$30# per day and for each dollar increase, #5# fewer cars are rented. What is the maximum possible income and the rent at which this maximizes?
• A rectangular page is to contain 16 square inches of print. The margins on each side are 1 inch. Find the dimensions of the page such that the least amount of paper is used?
• Find the length and width of a rectangle that has the given perimeter and a maximum area? Perimeter: 164 meters
• Find two positive numbers that satisfy the given requirements. The sum of the first number squared and the second number is 60 and the product is a maximum?
• Question #1b8d8
• 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#?
• Question #5cf1a
• The corners are removed from a sheet of paper that is #3# ft square. The sides are folded up to form an open square box. What is the maximum volume of the box?
• An arched window (an upper semi-circle and lower rectangle) has a total perimeter of #10 \ m#. What is the maximum area of the window?
• 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
• Solve this problem? It's so hard for me
• Explain Please?
• Can some one help with this one? I am new to derivatives.