The price-demand equation and the cost function for the production of table saws are given, respectively, by

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

Number of saws = x = 8400 - 36p

Total cost of production = C(x) = 60000 + 72x

Where p = price per saw.

**Solution**

(A) Price p as function of demand x

We get this by rearranging the terms in equation for x as follows:

x = 8400 - 36p

36p = 8400 - x

Therefore price as a function of x is represented by

P(x) = 8400/36 - x/36 = 300 - x/36

(B) Marginal cost

Marginal cost is represented by the slope of the cost curve C(x) = 60000 + 72x

This is equal to 72.

Therefor marginal cost is $72

(C) - 1. Revenue Function

Revenue is given by the function

R (x)= Price function * Volume = P(x)*x

= (300 - x/36)*x = 300x - x^2/36

(C) - 2. Marginal Revenue

Marginal revenue is represented by the slope of the revenue curve, R(x) = 300x - x^2/36

This slope is represented by Derivative of R(x)

Therefore:

Marginal Revenue Function = M(x) = R'(x) = 300 - x/18

(C) - 3. Profit function

Profit function = S(x) = Revenue function - Cost Function = R(x) - C(x)

= (300x - x^2/36) - (60000 + 72x)

= - x^2/36 + 228x - 60000

(C) - 4. Marginal Profit

Marginal profit is represented by the slope of the profit curve, S(x) = x^2/36 + 228x - 60000

Marginal Profit function = S'(x) = - x/18 + 228

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