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The Pear company sells pPhones. The cost to manufacture Iphones isC(x)= -21x^2+52000x + 20940dollars (this includes overhead costs and production costsfor each pPhone). If the company sells pPhones for the maximum price they can fetch, therevenue function will be R(x) = -29x^2+ 196000x dollars.How many pPhones should the Pear company produce and sell to maximimze profit? (Rememberthat profit-revenue-cost)

Answer :

ANSWER:

900

STEP-BY-STEP EXPLANATION:

Given the cost and revenue functions:

[tex]\begin{gathered} C\mleft(x\mright)=-21x^2+52000x+20940 \\ R\mleft(x\mright)=-29x^2+196000x \end{gathered}[/tex]

The profit is:

[tex]\begin{gathered} P(x)=R(x)-C(x) \\ P(x)=-29x^2+196000x-(-21x^2+52000x+20940) \\ P(x)=-29x^2+196000x+21x^2-52000x-20940 \\ P(x)=-8x^2+144000x-20940 \end{gathered}[/tex]

To maximize we derive the profit function and then set it equal to 0 to solve for x

[tex]\begin{gathered} P^{\prime}(x)=\frac{d}{dx}(-8x^2+144000x-20940) \\ P^{\prime}(x)=-16x+144000 \\ P^{\prime}(x)=0 \\ 0=-16x+144000 \\ 16x=14400 \\ x=\frac{14400}{16} \\ x=900 \end{gathered}[/tex]

To maximize profit, a total of 900 pPhones must be produced and sold.

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