Maths: Integral Calculus: Application of Integration in Economics and Commerce: Revenue functions from Marginal revenue functions

**Revenue functions from Marginal revenue
functions**

If *R* is the total revenue function when the output is *x*,
then marginal revenue *MR *=* dR/dx *Integrating with respect to â€˜* x *â€™ we get

Revenue Function, *R* =
âˆ« ( *MR* ) *dx* + *k*.

Where â€˜*k*â€™ is the constant of integration which can be
evaluated under given conditions, when *x* = 0, the total revenue *R*
= 0,

Demand Function, P=R/x, *x* â‰ 0

**Example 3.16**

For the marginal revenue function *MR* = 35 + 7*x* âˆ’ 3*x*^{2} ,
find the revenue function and demand function.

*Solution:*

**Example 3.17**

A firm has the marginal revenue function given by MR = where x is the output and a, b, c are constants. Show that the demand function is given by x =

*Solution:*

If â€˜*P*â€™ denotes the profit function, then Integrating both sides with respect to *x* gives , *P* = âˆ«( *MR* âˆ’ *MC* ) *dx* + *k*

Where *k* is the constant of integration. However if we are
given additional information, such as fixed cost or loss at zero level of
output, we can determine the constant *k*. Once *P* is known, it can
be maximum by using the concept of maxima and minima.

**Example 3.18**

The marginal cost *C *â€²* *(* x*) and marginal revenue *R* â€²
( *x*) are given by *C *â€²* *(* x*)* *=* *50* *+ x/50 *and R*â€²* *(* x*)=* *60* *. The fixed
cost is â‚¹200. Determine the
maximum profit.

*Solution:*

When quantity produced is zero, then the fixed cost is 200.

i.e. When *x *= 0,* c *= 200

*k*_{1 }=200

=5000 â€“ 2500 â€“ 200

=2300

Profit = â‚¹
2,300.

**Example 3.19**

The marginal cost and marginal revenue with respect to commodity
of a firm are given by *C* â€²
( *x*) = 8 + 6*x* and *R* â€² ( *x*)= 24. Find the total
Profit given that the total cost at zero output is zero.

*Solution:*

**Example 3.20**

The marginal revenue function (in thousand of rupees ) of a
commodity is 10 +
*e*^{âˆ’}^{0. 05x} Where *x* is the
number of units sold. Find the total revenue from the sale of 100 units (*e*^{âˆ’}^{5} = 0.0067)

*Solution:*

Given, Marginal revenue *R* â€² (
*x*) = 10 + *e*^{âˆ’}^{0 .05x}

Total revenue from sale of 100 units is

Total revenue = 1019.87 Ã— 1000

= â‚¹10,19,870

**Example 3.21**

The price of a machine is â‚¹5,00,000
with an estimated life of 12 years. The estimated salvage value is â‚¹30,000. The
machine can be rented at â‚¹72,000 per year. The present value of the rental
payment is calculated at 9% interest rate. Find out whether it is advisable to
rent the machine. (*e*^{âˆ’}^{1 .08} = 0.3396) .

*Solution:*

Cost of the machine =
5, 00, 000 âˆ’ 30, 000

=4, 70, 000

Hence it not advisable to rent the machine

It is better to buy the machine.

Given the inventory on hand *I* ( *x* ) and the unit
holding cost (*C*_{1}), the total inventory carrying
cost is *C*_{1 0}âˆ«^{T}_{0} *I* (*x* )*dx*, where *T*
is the time period under consideration.

**Example 3.22**

A company receives a shipment of 200 cars every 30 days. From
experience it is known that the inventory on hand is related to the number of
days. Since the last shipment, *I *(* x*)=* *200* *âˆ’* *0.2*x *. Find the daily holding cost for maintaining
inventory for 30 days if the* *daily holding cost is â‚¹3.5

*Solution:*

The amount of an annuity is the sum of all payments made plus all
interest accumulated. Let an annuity consist of equal payments of Rs. p and let
the interest rate of r percent annually be compounded continuously.

Amount of annuity after *N* payments *A* =

**Example 3.23**

Mr. Arul invests â‚¹10,000 in ABC Bank each year, which pays an
interest of 10% per annum compounded continuously for 5 years. How much amount
will there be after 5 years. (
*e*^{0 .5} =
1.6487)

*Solution:*

p = 10000, r = 0.1, N = 5

Suppose that p(t) is the annual consumption of a natural resource
in year t. If the consumption of the resource is growing exponentially at
growth rate k, then the total consumption of the resource after T years is
given by

Where *p*_{0} is the initial annual consumption at
time *t* = 0.

**Example 3.24**

In year 2000 world gold production was 2547 metric tons and it was
growing exponentially at the rate of 0.6% per year. If the growth continues at
this rate, how many tons of gold will be produced from 2000 to 2013? [e^{0.078}
= 1.0811)

*Solution:*

Annual
consumption at time t = 0 (In the year 2000) = p_{0} = 2547 metric
ton.

Total production of Gold from 2000 to 2013 =

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12th Business Maths and Statistics : Chapter 3 : Integral Calculus - II : Integration: Revenue functions from Marginal revenue functions | Example Solved Problems with Answer, Solution, Formula

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