Section 4.7 Optimization Problems . Area. Find the dimensions of the rectangular field of maximum area that can be made from 300 m of fencing material. 25 m. 50 m. 125 m. 100 m. 10 m. 75 m. 140 m. 75 m. Optimization Problem.
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Area Find the dimensions of the rectangular field of maximum area that can be made from 300 m of fencing material. 25 m 50 m 125 m 100 m 10 m 75 m 140 m 75 m
Optimization Problem 1. Assign variables and draw a figure/diagram, if possible. 2.Write the condition/constrain for the situation 3.Write an equation that relates the quantity to be optimized to other variables. 4.Use the conditions of the problem to write the quantity to be optimized as a function of ONE variable. 5.Find the feasible domain of that function 6.Find the absolute extremaof the function on the specified domain.
There must be a local maximum here, since the endpoints are minimums. A Classic Problem You have 40 feet of fence to enclose a rectangular garden along the side of a barn. What is the maximum area that you can enclose?
A Classic Problem You have 40 feet of fence to enclose a rectangular garden along the side of a barn. What is the maximum area that you can enclose?
Example : What dimensions for a one liter cylindrical can will use the least amount of material? Motor Oil We can minimize the material by minimizing the area. We need another equation that relates r and h: area of ends lateral area
Example : What dimensions for a one liter cylindrical can will use the least amount of material? area of ends lateral area
Packing Design A television manufacturing firm needs to design an open-topped box with a square base. The box must hold 32 cu. in. Find the dimensions of a box that can be built with the minimum amount of materials.
Minimum Cost A fence must be build to enclose a rectangular area of 20,000 sq. ft. Fencing material costs $2.50 per foot for the two sides facing north and south, and $3.20 per foot for the other two sides. Find the cost of the least expensive fence.
Minimum length Two posts, one 12 feet high and the other 28 feet high, stand 30 feet apart. They are to be stayed by two wires, attached to a single stake, running from ground level to the top of each post. Where should the stake be placed to use the least amount of wire? Note: This problem involves a lot of algebra. You can use technology to help solve the problem. http://www.wolframalpha.com