Sec 4.6: Applied Optimization EXAMPLE 1 An open-top box is to be made by cutting small congruent...
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Sec 4.6: Applied Optimization EXAMPLE 1 An open-top box is to be made by cutting small congruent squares from the corners of a 12-in.- by-12-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?
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Solving Applied Optimization Problems 1. Read the problem. Read the problem until you understand it. What is given? What is the unknown quantity to be optimized? 2. Draw a picture. Label any part that may be important to the problem. 3. Introduce variables. List every relation in the picture and in the problem as an equation or algebraic expression, and identify the unknown variable. Sec 4.6: Applied Optimization 4. Write an equation for the unknown quantity (maximize or minimize). If you can, express the unknown as a function of a single variable or in two equations in two unknowns. This may require considerable manipulation. 5. Find the domain of the single variable function. The possible value of x in the problem 6. Test the critical points and endpoints in the domain of the unknown. Use your knowledge from section to find the global maximum or global minimum The cutout squares should be 2 in. on a side. EXAMPLE 1 An open-top box is to be made by cutting small congruent squares from the corners of a 12-in.-by-12-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? Criticals are x=2, 6
Transcript of Sec 4.6: Applied Optimization EXAMPLE 1 An open-top box is to be made by cutting small congruent...
Sec 4.6: Applied Optimization
EXAMPLE 1 An open-top box is to be made by cutting small congruent squares from the corners of a 12-in.-by-12-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?
Sec 4.6: Applied Optimization
EXAMPLE 1 An open-top box is to be made by cutting small congruent squares from the corners of a 12-in.-by-12-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?
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Solving Applied Optimization Problems
1. Read the problem. Read the problem until you understand it. What is given?What is the unknown quantity to be optimized?
2. Draw a picture. Label any part that may be important to the problem.
3. Introduce variables. List every relation in the picture and in the problem as an equation or algebraic expression, and identify the unknown variable.
Sec 4.6: Applied Optimization
) ( 212)( sidebasexs
heightxh
4. Write an equation for the unknown quantity (maximize or minimize). If you can, express the unknown as a function of a single variable or in two equations in two unknowns. This may require considerable manipulation.
5. Find the domain of the single variable function. The possible value of x in the problem
6. Test the critical points and endpoints in the domain of the unknown. Use your knowledge from section 4.1 -4.4 to find the global maximum or global minimum
)( )212( 2 volumexxv
60 x
]6,0[ interval on the)212()(
max golbal theFind2xxxv
x212
x
)612)(212( xxdxdv
60 200 128
The cutout squares should be 2 in. on a side.
EXAMPLE 1 An open-top box is to be made by cutting small congruent squares from the corners of a 12-in.-by-12-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?
Criticals are x=2, 6
Solving Applied Optimization Problems
1. Read the problem.2. Draw a picture
3. Introduce variables.
4. Write an equation for the unknown quantity (maximize or minimize). .
5. Find the domain of the single variable function.
6. Test the critical points and endpoints in the domain of the unknown.
Sec 4.6: Applied OptimizationEXAMPLE 1 A farmer has 2400 ft of fencing and wants to fence off a rectangular field that borders a straight river. He needs no fence along the river. What are the dimensions of the field that has the largest area?
) (sec 22400) ( sideondx-
sideonex
)( )22400( AreaxxA
12000 x
]1200,0[ interval on the)22400()(
max golbal theFindxxxA
222400 xxdxdA
12000 600
0A 000,720A
The dimensions is 600X1200
Criticals are x= 600
0A
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Sec 4.6: Applied Optimization
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Sec 4.6: Applied Optimization
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xhxasurfaceAre 42
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240002 4x
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xxS 160002
2160002xdx
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200 xdxdS
120020 Sx xx
h
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Sec 4.6: Applied Optimization
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Sec 4.6: Applied Optimization
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Sec 4.6: Applied Optimization
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Sec 4.6: Applied Optimization
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Sec 4.6: Applied Optimization
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