x = 8400 – 36p and C(x) = 60000 + 72x, The price-demand equation and the cost function for the production of table saws are given, respectively, by x = 8400 – 36p and C(x) = 60000 + 72x, where...

x = 8400 – 36p and C(x) = 60000 + 72x, 

The price-demand equation and the cost function for the production of table saws are given, respectively, by
x = 8400 – 36p and C(x) = 60000 + 72x, where x is the number of saws that can be sold a price of $p per saw and C(x) is the
total cost (in dollars) of producing a saw. (A) Express the price p as a function of the demand x. (B) Find the marginal cost.
(C) Find the revenue function, marginal revenue, profit function, and marginal profit.

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sciencesolve | Teacher | (Level 3) Educator Emeritus

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A) You need to express the price as a function of demand, hence, you need to find the inverse of the function `x = 8400 -36p`  such that:

`x = 8400 -36p =gt x - 8400 = -36p`  (you need to isolate p)

`p = (x - 8400)/(-36) =gt p(x) = (8400 - x)/(36)`

Hence, evaluating the price as a function of demand yields `p(x) = (8400 - x)/(36).` B) You need to find the marginal cost, hence, you should differentiate the cost function with respect to x such that:

`C'(x) = (60000 + 72x)' =gt C'(x) = 72`

Hence, evaluating the marginal cost yields C'(x) = 72.

C) You should find the revenue function such that:

R(x) = x*p(x)

The function p(x) is found at A), hence `R(x) = x*(8400 - x)/(36)`

You may find the marginal revenue, differentiating the revenue function with respect to x such that:

`R'(x) = x'*(8400 - x)/(36) + x*((8400 - x)/(36))'`

`R'(x) = (8400 - x)/(36) - x/36`

`R'(x) = (8400 - 2x)/36 =gt R'(x) = (4200 - x)/18`

You need to find the profit function such that:

`P(x) = R(x)- C(x)`

`P(x) = x*(8400 - x)/(36) - 60000- 72x`

`P(x) = 8400x - x^2 - 60000*36 - 72*36x`

`P(x) = -x^2 + 5808x - 2160000`

You may find the marginal profit differentiating the profit function with respect to x such that:

`P'(x) = -2x + 5808`

Hence, evaluating the revenue function, the marginal revenue, the profit function and the marginal profit yields `R(x) = x*(8400 - x)/(36) ; R'(x) = (4200 - x)/18 ; P(x) = -x^2 + 5808x - 2160000 ; P'(x) = -2x + 5808.`

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