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.

Expert Answers

An illustration of the letter 'A' in a speech bubbles

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...

See
This Answer Now

Start your 48-hour free trial to unlock this answer and thousands more. Enjoy eNotes ad-free and cancel anytime.

Get 48 Hours Free Access

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.`

Approved by eNotes Editorial Team