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This is section 3.7 problem 60: a clothing manufacturer has the cost function c(x)=1200+30x+0.5x2 , (in dollars), 0≤ x≤ 250 , where x is the number of suits produced each week. the revenue function for selling x suits is given by r(x)=120x , (in dollars). in order to achieve the maximum profit each week, suits per week must be produced and sold to achieve the maximum profit of $ . hint: follow example 1.

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Answer:

90 suits per week must be produced and sold to achieve the maximum profit of $2,850.

Explanation:

The profit function is given by the revenue function minus the cost function:

[tex]P(x) = R(x) - C(x)\\P(x)=120x -1200-30x-0.5x^2[/tex]

The number of suits, x, for which the derivate of the profit funtion is zero, is the production volume that maximizes profit:

[tex]P'(x)=0=120-30-x\\x=90\ suits[/tex]

The profit generated by producing 90 suits is:

[tex]P(90)=120*90 -1200-30*90-0.5*90^2\\P(90) = \$2,850[/tex]

Therefore, 90 suits per week must be produced and sold to achieve the maximum profit of $2,850.

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