# If the cost function and demand curve for a certain product are `C (x)= 60x + 7200` and `P(x)= 300 -2x` respectively: Find a. the total revenue function b. the marginal revenue function c. the...

If the cost function and demand curve for a certain product are `C (x)= 60x + 7200` and `P(x)= 300 -2x` respectively:

Find

a. the total revenue function

b. the marginal revenue function

c. the marginal cost function

d. point(s) at which there is breakeven.

e. the value of x at which marginal revenue equal to marginal cost. Show all work where required.

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(a.) Total revenue function

Apply the formula,

Revenue = Price per unit `xx` Number of units sold

Take note that price per unit is the same as demand function. So,

`R(x) = (300-2x)*x`

`R(x)=300x-2x^2`

**Hence, the revenue function is `R(x) = 300x-2x^2` .**

(b.) Marginal revenue function

The formula for marginal revenue function is:

Marginal Revenue= R'(x)

So, take the derivative of R(x).

`R'(x) =(300x-2x^2)'`

`R'(x)=300-4x`

**Thus, the marginal revenue function is `R'(x) =300-4x` .**

(c.) Marginal cost function

To determine the marginal cost function, take the derivative of C(x).

Marginal Cost = C'(x)

`C'(x)=(60x + 7200)'`

`C'(x)=60`

**Therefore, the marginal cost function is `C'(x) =60` .**

d. Point(s) at which there is break-even.

To determine the break even point, set the cost and revenue functions equal to each other.

`C(x)=R(x)`

`60x+7200=300x-2x^2`

Then, set one side equal to zero.

`2x^2+60x-300x+7200=0`

`2x^2-240x +7200=0`

Simplify the equation and factor.

`x^2-120x+3600=0`

`(x-60)(x-60)=0`

And, set the factor equal to zero to get the value of x.

`x - 60 = 0`

`x=60`

Next, plug-in the value of x to either cost or revenue function.

`C(x) =60x+7200`

`C(60)=60(60)+7200`

`C(60)=10800`

**Hence, the break-even point is (60,10800).**

e. The value of x at which marginal revenue equal to marginal cost.

Set R'(x) and C'(x) equal.

`R'(x)=C'(x)`

`300-4x =60`

And, solve for x.

`-4x=60-300`

`-4x=-240`

`x=60`

**Thus, the marginal revenue and marginal cost are equal when x=60.**

a) Total Revenue(x) = Quantity(x) * Price(x)

= x* ( 300-2x)

b) Marginal Revenue(x) = d/dx( Total Revenue)

=`dy/dxx*(300-2x)`

` `

= 300-2x -2x

= 300-4x

c) Marginal Cost(x) = d/dx( Cost(x) )

= d/dx( 60x + 7200 )

= 60

d) For Breakeven , Marginal Revenue = Marginal Cost

`:.` 300 - 4x = 60

x = 60

e) At quantity x= 60, Marginal revenue will be same as Marginal cost.