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The cost, in dollars, to produce vats of ice cream is C(x) = 2x+4. When selling them to ice creamshops, the price-demand function, in dollars per vat, is p(x) = 116 - 32Find the profit function.P(x)How many vats of ice cream need to be sold to maximize the profit.vats of ice creamFind the maximum profit.dollars

The cost in dollars to produce vats of ice cream is Cx 2x4 When selling them to ice creamshops the pricedemand function in dollars per vat is px 116 32Find the class=

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SOLUTION

From the question,

The cost is given by the function

[tex]C(x)=2x+4[/tex]

And the price demand function is given by the function

[tex]p(x)=116-3x[/tex]

Now profit is calculated as

[tex]P(x)=xp(x)-C(x)[/tex]

So we have

[tex]\begin{gathered} P(x)=xp(x)-C(x) \\ P(x)=x(116-3x)-(2x+4) \\ =116x-3x^2-2x-4 \\ =-3x^2-114x-4 \end{gathered}[/tex]

Hence the profit function is

[tex]P(x)=-3x^2-114x-4[/tex]

At maximum profit, the derivative of the function for profit is equal to zero, we have

[tex]\begin{gathered} P(x)=-3x^2-114x-4 \\ P^{\prime}(x)=-6x^{}-114 \\ -6x^{}-114=0 \\ 6x=-114 \\ x=\frac{-114}{6} \\ x=-19 \end{gathered}[/tex]

So we can see that the answer to that is -19

The maximum profit becomes ,

we substitute x for -19, we have

[tex]\begin{gathered} P(-19)=-3x^2-114x-4 \\ P(-19)=-3(-19)^2-114(-19)-4 \\ =-1,083+2,166-4 \\ =1,079 \end{gathered}[/tex]

Hence the maximum profit is $1,079

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