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Suppose that the weekly​ profit, in​ dollars, of producing and selling x cars is P (x) = negative 0.007 x cubed - 0.2 x ^2 + 1000 x - 1100,and currently 60 cars are produced and sold weekly. Use P (60) and the marginal profit to estimate the weekly profit of producing and selling 62 cars. Round to the nearest dollar.

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Answer:

  • The weekly profit "P" of producing and selling "x" cars, is given by the expression [tex]P(x)= -0.007x^3-0.2x^2+1000x-1100[/tex].
  • If the amount of cars produced and sold are x=60, then the profit linked to those 60 cars will be given by [tex]P(60)= -0.007(60)^3-0.2(60)^2+1,000(60)-1,100[/tex], where the amount of cars "x" has been replaced by 60. This equals [tex]P(60)=56,668[/tex].
  • Marginal profit is defined as the change in profits when the amount of cars produced and sold change in a unit. Matematically, it equals the derivative of P(x) in terms of x: [tex]\frac{\partial P(x)}{\partial x}= -0.021x^2-0.4x+1,000[/tex] .
  • If the amount of cars produced and sold change in two units, then the marginal profit (or additional profit) obtained by producing and selling these two additional cars is [tex]-0.021(2)^2-0.4(2)+1,000[/tex] ≈ [tex]1,000[/tex].
  • Then, producing and selling 62 cars would produce a benefit of [tex]\$56,668+\$1,000= \$57,668[/tex].

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