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:
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.
(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`
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).
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)'`
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.
Then, set one side equal to zero.
Simplify the equation and factor.
And, set the factor equal to zero to get the value of x.
`x - 60 = 0`
Next, plug-in the value of x to either cost or revenue function.
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.
And, solve for x.
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)
= 300-2x -2x
c) Marginal Cost(x) = d/dx( Cost(x) )
= d/dx( 60x + 7200 )
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.