16bf10lec3 - Math 16b Thomas Scanlon Autumn 2010 Thomas...

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Unformatted text preview: Math 16b Thomas Scanlon Autumn 2010 Thomas Scanlon Math 16b An example Consider a rectangle one of whose side lengths is x and the other y The area of this rectangle may be given by the formula A ( x , y ) := xy . It is not possible to choose x and y so as to maximize this area, but if we impose a constraint between x and y there may be a best choice of x and y . Thomas Scanlon Math 16b Example, continued Let us suppose that the material for the red (left vertical) side of this rectangle costs $ 7 per foot while the material for all the other sides costs $ 5 per foot and that we have a budget of $ 600. What are the dimensions which will maximize the area subject to this budget? Thomas Scanlon Math 16b Solution We have the relation 600 = 7 y + 5 y + 5 x + 5 x = 12 y + 10 x Solving for y in terms of x , we have y = 50- 5 6 x . Substituting this expression for y into the formula for the area we see the area as a function of a single variable α ( x ) := A ( x , 50- 5 6 x ) = x ( 50- 5 6 x...
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