Home | | Business Maths 12th Std | Integration: Cost functions from marginal cost functions

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

Chapter: 12th Business Maths and Statistics : Chapter 3 : Integral Calculus - II

Integration: Cost functions from marginal cost functions

If C is the cost of producing an output x, then marginal cost function MC = dc/dx

Cost functions from marginal cost functions

If C is the cost of producing an output x, then marginal cost function MC = dc/dx

Using integration, as the reverse process of differentiation, we obtain,

Cost function C = ( MC ) dx + k

Where k is the constant of integration which is to be evaluated,

Average cost function AC = C/X, x 0

 

Example 3.9

The marginal cost function of manufacturing x shoes is 6 + 10x 6x2 . The cost producing a pair of shoes is ₹12. Find the total and average cost function.

Solution:

Given,

Marginal cost MC = 6 + 10x 6x2

C = MC dx + k

= ∫( 6 + 10x 6x2 ) dx + k

= 6x + 5x22x 3 + k (1)

When x= 2 , C = 12 (given)

12 = 12 + 20 16 + k

k=-4

C = 6x + 5x2 2x3 4


 

Example 3.10

A company has determined that the marginal cost function for a product of a particular commodity is given by MC = 125 + 10x x2/9 where C rupees is the cost of 9 producing x units of the commodity. If the fixed cost is ₹250 what is the cost of producing 15 units.

Solution:


 

Example 3.11

The marginal cost function MC = 2 + 5ex (i) Find C if C (0)=100 (ii) Find AC.

Solution:


 

Rate of growth or sale

If the rate of growth or sale of a function is a known function of t say f(t) where t is a time measure, then total growth (or) sale of a product over a time period t is given by,

Total sale = 

 

Example 3.12

The rate of new product is given by f (x) = 100 90 ex where x is the number of days the product is on the market. Find the total sale during the first four days. (e–4=0.018)

Solution:


 

Example 3.13

A company produces 50,000 units per week with 200 workers. The rate of change of productions with respect to the change in the number of additional labour x is represented as 300 5x 2/3 . If 64 additional labours are employed, find out the additional number of units, the company can produce.

Solution:

Let p be the additional product produced for additional of x labour,


The number of additional units produced 16128

Total number of units produced by 264 workers

50,000 + 16,128 = 66128 units

 

Example 3.14

The rate of change of sales of a company after an advertisement campaign is represented as, f (t ) = 3000e 0.3t where t represents the number of months after the advertisement. Find out the total cumulative sales after 4 months and the sales during the fifth month. Also find out the total sales due to the advertisement campaign e 1 . 2 = 0.3012, e1.5 = 0.2231 .

Solution:


 

Example 3.15

The price of a machine is 6,40,000 if the rate of cost saving is represented by the function f(t) = 20,000 t. Find out the number of years required to recoup the cost of the function.

Solution:

Saving Cost          S(t) = t0 20000t dt

= 10000 t2

To recoup the total price,

10000 t2 = 640000

t2 = 64

t = 8

When t = 8 years, one can recoup the price.


Tags : Example Solved Problems with Answer, Solution, Formula , 12th Business Maths and Statistics : Chapter 3 : Integral Calculus - II
Study Material, Lecturing Notes, Assignment, Reference, Wiki description explanation, brief detail
12th Business Maths and Statistics : Chapter 3 : Integral Calculus - II : Integration: Cost functions from marginal cost functions | Example Solved Problems with Answer, Solution, Formula

Related Topics

12th Business Maths and Statistics : Chapter 3 : Integral Calculus - II


Privacy Policy, Terms and Conditions, DMCA Policy and Compliant

Copyright © 2018-2023 BrainKart.com; All Rights Reserved. Developed by Therithal info, Chennai.