Chapter 7.5, Problem 4CP

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10th Edition

Ron Larson

ISBN: 9781305860919

Textbook Problem

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Find the prices that will yield a maximum profit for the products in Example 4 when the costs of producing the two products are $0.75 and $0.50 per unit, respectively.

To determine

**To calculate:** The unit prices that will yield a maximum profit for the products, which are substitute, with the respective demand functions given by,

Where

Explanation

**Given Information:**

The demand function for the product are given by,

Where

The costs of producing the two products are dollar 0.75 and dollar 0.50 per unit respectively.

**Formula used:**

The demand functions for any pairs of substance products produced by a company is,

The cost function of the pair of substitute products for the company is,

The reverse function of the pair of substitute products for the company is,

and the profit function for the company is,

Where

The following procedure is used to optimise the profit function:

Step 1: Write the primary equation for the quantities that are to be optimised, there the primary equations is the profit function.

Step 2: Express the primary equation, in terms of the quantities that are to the optimised, with help of secondary tertiary and other functions.

Step 3: Find the first partial derivatives of the primary equation with respect to each quantity that are to be optimised, separate by.

Step 4: Set the first partial derivatives to zero find the critical values of the required quantities.

Step 5: Find second partial derivatives to check if the critical values, obtained at step 4, helps to satisfy the required condition.

Step 6: If the required condition is satisfied, then use the critical values to find further results. Else, state that the optimisation cannot be reached.

**Calculation:**

The primary equation is the profit function,

Where P is the total profit, R is the total revenue and c is the total cost.

The secondary equation in the given demand function as,

Where

The tertiary equation is the cost function here, with the given respective per unit costs as shorn below.

Where *C*.

The quaternary equation is the revenge function here, with R as total revenue as shown below :

Where

Put form the demand function in the tertiary equation to get,

Put

Put the expressions obtained for C and R in the primary equation to get5 the expressed primary equations

Sect-7.1 P-1CPSect-7.1 P-2CPSect-7.1 P-3CPSect-7.1 P-4CPSect-7.1 P-5CPSect-7.1 P-6CPSect-7.1 P-7CPSect-7.1 P-1SWUSect-7.1 P-2SWUSect-7.1 P-3SWU

Sect-7.1 P-4SWUSect-7.1 P-5SWUSect-7.1 P-6SWUSect-7.1 P-7SWUSect-7.1 P-8SWUSect-7.1 P-1ESect-7.1 P-2ESect-7.1 P-3ESect-7.1 P-4ESect-7.1 P-5ESect-7.1 P-6ESect-7.1 P-7ESect-7.1 P-8ESect-7.1 P-9ESect-7.1 P-10ESect-7.1 P-11ESect-7.1 P-12ESect-7.1 P-13ESect-7.1 P-14ESect-7.1 P-15ESect-7.1 P-16ESect-7.1 P-17ESect-7.1 P-18ESect-7.1 P-19ESect-7.1 P-20ESect-7.1 P-21ESect-7.1 P-22ESect-7.1 P-23ESect-7.1 P-24ESect-7.1 P-25ESect-7.1 P-26ESect-7.1 P-27ESect-7.1 P-28ESect-7.1 P-29ESect-7.1 P-30ESect-7.1 P-31ESect-7.1 P-32ESect-7.1 P-33ESect-7.1 P-34ESect-7.1 P-35ESect-7.1 P-36ESect-7.1 P-37ESect-7.1 P-38ESect-7.1 P-39ESect-7.1 P-40ESect-7.1 P-41ESect-7.1 P-42ESect-7.1 P-43ESect-7.1 P-44ESect-7.2 P-1CPSect-7.2 P-2CPSect-7.2 P-3CPSect-7.2 P-1SWUSect-7.2 P-2SWUSect-7.2 P-3SWUSect-7.2 P-4SWUSect-7.2 P-5SWUSect-7.2 P-6SWUSect-7.2 P-1ESect-7.2 P-2ESect-7.2 P-3ESect-7.2 P-4ESect-7.2 P-5ESect-7.2 P-6ESect-7.2 P-7ESect-7.2 P-8ESect-7.2 P-9ESect-7.2 P-10ESect-7.2 P-11ESect-7.2 P-12ESect-7.2 P-13ESect-7.2 P-14ESect-7.2 P-15ESect-7.2 P-16ESect-7.2 P-17ESect-7.2 P-18ESect-7.2 P-19ESect-7.2 P-20ESect-7.2 P-21ESect-7.2 P-22ESect-7.2 P-23ESect-7.2 P-24ESect-7.2 P-25ESect-7.2 P-26ESect-7.2 P-27ESect-7.2 P-28ESect-7.2 P-35ESect-7.2 P-36ESect-7.2 P-37ESect-7.2 P-38ESect-7.2 P-39ESect-7.2 P-40ESect-7.2 P-41ESect-7.2 P-42ESect-7.2 P-43ESect-7.2 P-44ESect-7.2 P-45ESect-7.2 P-46ESect-7.2 P-47ESect-7.2 P-48ESect-7.2 P-49ESect-7.2 P-50ESect-7.2 P-51ESect-7.3 P-1CPSect-7.3 P-2CPSect-7.3 P-3CPSect-7.3 P-4CPSect-7.3 P-5CPSect-7.3 P-1SWUSect-7.3 P-2SWUSect-7.3 P-3SWUSect-7.3 P-4SWUSect-7.3 P-5SWUSect-7.3 P-6SWUSect-7.3 P-7SWUSect-7.3 P-8SWUSect-7.3 P-9SWUSect-7.3 P-10SWUSect-7.3 P-1ESect-7.3 P-2ESect-7.3 P-3ESect-7.3 P-4ESect-7.3 P-5ESect-7.3 P-6ESect-7.3 P-7ESect-7.3 P-8ESect-7.3 P-9ESect-7.3 P-10ESect-7.3 P-11ESect-7.3 P-12ESect-7.3 P-13ESect-7.3 P-14ESect-7.3 P-15ESect-7.3 P-16ESect-7.3 P-17ESect-7.3 P-18ESect-7.3 P-19ESect-7.3 P-20ESect-7.3 P-21ESect-7.3 P-22ESect-7.3 P-23ESect-7.3 P-24ESect-7.3 P-25ESect-7.3 P-26ESect-7.3 P-27ESect-7.3 P-28ESect-7.3 P-29ESect-7.3 P-30ESect-7.3 P-31ESect-7.3 P-32ESect-7.3 P-33ESect-7.3 P-34ESect-7.3 P-35ESect-7.3 P-36ESect-7.3 P-37ESect-7.3 P-38ESect-7.3 P-39ESect-7.3 P-40ESect-7.3 P-41ESect-7.3 P-42ESect-7.3 P-43ESect-7.3 P-44ESect-7.3 P-45ESect-7.3 P-46ESect-7.3 P-47ESect-7.3 P-48ESect-7.3 P-49ESect-7.3 P-50ESect-7.3 P-51ESect-7.3 P-53ESect-7.3 P-54ESect-7.3 P-55ESect-7.4 P-1CPSect-7.4 P-2CPSect-7.4 P-3CPSect-7.4 P-4CPSect-7.4 P-5CPSect-7.4 P-6CPSect-7.4 P-7CPSect-7.4 P-1SWUSect-7.4 P-2SWUSect-7.4 P-3SWUSect-7.4 P-4SWUSect-7.4 P-5SWUSect-7.4 P-6SWUSect-7.4 P-7SWUSect-7.4 P-8SWUSect-7.4 P-9SWUSect-7.4 P-10SWUSect-7.4 P-1ESect-7.4 P-2ESect-7.4 P-3ESect-7.4 P-4ESect-7.4 P-5ESect-7.4 P-6ESect-7.4 P-7ESect-7.4 P-8ESect-7.4 P-9ESect-7.4 P-10ESect-7.4 P-11ESect-7.4 P-12ESect-7.4 P-13ESect-7.4 P-14ESect-7.4 P-15ESect-7.4 P-16ESect-7.4 P-17ESect-7.4 P-18ESect-7.4 P-19ESect-7.4 P-20ESect-7.4 P-21ESect-7.4 P-22ESect-7.4 P-23ESect-7.4 P-24ESect-7.4 P-25ESect-7.4 P-26ESect-7.4 P-27ESect-7.4 P-28ESect-7.4 P-29ESect-7.4 P-30ESect-7.4 P-31ESect-7.4 P-32ESect-7.4 P-33ESect-7.4 P-34ESect-7.4 P-35ESect-7.4 P-36ESect-7.4 P-37ESect-7.4 P-38ESect-7.4 P-39ESect-7.4 P-40ESect-7.4 P-41ESect-7.4 P-42ESect-7.4 P-43ESect-7.4 P-44ESect-7.4 P-45ESect-7.4 P-46ESect-7.4 P-47ESect-7.4 P-48ESect-7.4 P-49ESect-7.4 P-50ESect-7.4 P-51ESect-7.4 P-52ESect-7.4 P-53ESect-7.4 P-54ESect-7.4 P-55ESect-7.4 P-56ESect-7.4 P-57ESect-7.4 P-58ESect-7.4 P-59ESect-7.4 P-60ESect-7.4 P-61ESect-7.4 P-62ESect-7.4 P-63ESect-7.4 P-64ESect-7.4 P-65ESect-7.4 P-66ESect-7.4 P-67ESect-7.4 P-68ESect-7.4 P-69ESect-7.4 P-71ESect-7.4 P-72ESect-7.4 P-73ESect-7.5 P-1CPSect-7.5 P-2CPSect-7.5 P-3CPSect-7.5 P-4CPSect-7.5 P-5CPSect-7.5 P-1SWUSect-7.5 P-2SWUSect-7.5 P-3SWUSect-7.5 P-4SWUSect-7.5 P-5SWUSect-7.5 P-6SWUSect-7.5 P-7SWUSect-7.5 P-8SWUSect-7.5 P-9SWUSect-7.5 P-10SWUSect-7.5 P-11SWUSect-7.5 P-12SWUSect-7.5 P-13SWUSect-7.5 P-14SWUSect-7.5 P-1ESect-7.5 P-2ESect-7.5 P-3ESect-7.5 P-4ESect-7.5 P-5ESect-7.5 P-6ESect-7.5 P-7ESect-7.5 P-8ESect-7.5 P-9ESect-7.5 P-10ESect-7.5 P-11ESect-7.5 P-12ESect-7.5 P-13ESect-7.5 P-14ESect-7.5 P-15ESect-7.5 P-16ESect-7.5 P-17ESect-7.5 P-18ESect-7.5 P-19ESect-7.5 P-20ESect-7.5 P-21ESect-7.5 P-22ESect-7.5 P-23ESect-7.5 P-24ESect-7.5 P-25ESect-7.5 P-26ESect-7.5 P-27ESect-7.5 P-28ESect-7.5 P-29ESect-7.5 P-30ESect-7.5 P-31ESect-7.5 P-32ESect-7.5 P-33ESect-7.5 P-34ESect-7.5 P-35ESect-7.5 P-36ESect-7.5 P-37ESect-7.5 P-38ESect-7.5 P-39ESect-7.5 P-40ESect-7.5 P-41ESect-7.5 P-42ESect-7.5 P-43ESect-7.5 P-44ESect-7.5 P-45ESect-7.5 P-46ESect-7.5 P-47ESect-7.5 P-48ESect-7.5 P-49ESect-7.5 P-50ESect-7.5 P-51ESect-7.5 P-53ESect-7.5 P-54ESect-7.5 P-55ESect-7.5 P-56ESect-7.5 P-1QYSect-7.5 P-2QYSect-7.5 P-3QYSect-7.5 P-4QYSect-7.5 P-5QYSect-7.5 P-6QYSect-7.5 P-7QYSect-7.5 P-8QYSect-7.5 P-9QYSect-7.5 P-10QYSect-7.5 P-11QYSect-7.5 P-12QYSect-7.5 P-13QYSect-7.5 P-14QYSect-7.5 P-15QYSect-7.5 P-16QYSect-7.5 P-17QYSect-7.5 P-18QYSect-7.5 P-19QYSect-7.5 P-20QYSect-7.5 P-21QYSect-7.5 P-22QYSect-7.5 P-23QYSect-7.5 P-24QYSect-7.5 P-25QYSect-7.5 P-26QYSect-7.6 P-1CPSect-7.6 P-2CPSect-7.6 P-3CPSect-7.6 P-4CPSect-7.6 P-1SWUSect-7.6 P-2SWUSect-7.6 P-3SWUSect-7.6 P-4SWUSect-7.6 P-5SWUSect-7.6 P-6SWUSect-7.6 P-7SWUSect-7.6 P-8SWUSect-7.6 P-9SWUSect-7.6 P-10SWUSect-7.6 P-1ESect-7.6 P-2ESect-7.6 P-3ESect-7.6 P-4ESect-7.6 P-5ESect-7.6 P-6ESect-7.6 P-7ESect-7.6 P-8ESect-7.6 P-9ESect-7.6 P-10ESect-7.6 P-11ESect-7.6 P-12ESect-7.6 P-13ESect-7.6 P-14ESect-7.6 P-15ESect-7.6 P-16ESect-7.6 P-17ESect-7.6 P-18ESect-7.6 P-19ESect-7.6 P-20ESect-7.6 P-21ESect-7.6 P-22ESect-7.6 P-23ESect-7.6 P-24ESect-7.6 P-25ESect-7.6 P-26ESect-7.6 P-27ESect-7.6 P-28ESect-7.6 P-29ESect-7.6 P-30ESect-7.6 P-31ESect-7.6 P-32ESect-7.6 P-33ESect-7.6 P-34ESect-7.6 P-35ESect-7.6 P-36ESect-7.6 P-37ESect-7.6 P-38ESect-7.6 P-39ESect-7.6 P-40ESect-7.6 P-41ESect-7.6 P-42ESect-7.6 P-43ESect-7.6 P-44ESect-7.6 P-45ESect-7.7 P-1CPSect-7.7 P-2CPSect-7.7 P-3CPSect-7.7 P-1SWUSect-7.7 P-2SWUSect-7.7 P-3SWUSect-7.7 P-4SWUSect-7.7 P-5SWUSect-7.7 P-6SWUSect-7.7 P-7SWUSect-7.7 P-8SWUSect-7.7 P-1ESect-7.7 P-2ESect-7.7 P-3ESect-7.7 P-4ESect-7.7 P-5ESect-7.7 P-6ESect-7.7 P-7ESect-7.7 P-8ESect-7.7 P-9ESect-7.7 P-10ESect-7.7 P-11ESect-7.7 P-12ESect-7.7 P-13ESect-7.7 P-14ESect-7.7 P-15ESect-7.7 P-16ESect-7.7 P-17ESect-7.7 P-18ESect-7.7 P-19ESect-7.7 P-20ESect-7.7 P-21ESect-7.7 P-22ESect-7.7 P-23ESect-7.7 P-24ESect-7.7 P-25ESect-7.7 P-26ESect-7.7 P-27ESect-7.7 P-28ESect-7.7 P-29ESect-7.7 P-30ESect-7.8 P-1CPSect-7.8 P-2CPSect-7.8 P-3CPSect-7.8 P-4CPSect-7.8 P-5CPSect-7.8 P-6CPSect-7.8 P-1SWUSect-7.8 P-2SWUSect-7.8 P-3SWUSect-7.8 P-4SWUSect-7.8 P-5SWUSect-7.8 P-6SWUSect-7.8 P-7SWUSect-7.8 P-8SWUSect-7.8 P-9SWUSect-7.8 P-10SWUSect-7.8 P-11SWUSect-7.8 P-12SWUSect-7.8 P-13SWUSect-7.8 P-14SWUSect-7.8 P-15SWUSect-7.8 P-16SWUSect-7.8 P-1ESect-7.8 P-2ESect-7.8 P-3ESect-7.8 P-4ESect-7.8 P-5ESect-7.8 P-6ESect-7.8 P-7ESect-7.8 P-8ESect-7.8 P-9ESect-7.8 P-10ESect-7.8 P-11ESect-7.8 P-12ESect-7.8 P-13ESect-7.8 P-14ESect-7.8 P-15ESect-7.8 P-16ESect-7.8 P-17ESect-7.8 P-18ESect-7.8 P-19ESect-7.8 P-20ESect-7.8 P-21ESect-7.8 P-22ESect-7.8 P-23ESect-7.8 P-24ESect-7.8 P-25ESect-7.8 P-26ESect-7.8 P-27ESect-7.8 P-28ESect-7.8 P-29ESect-7.8 P-30ESect-7.8 P-31ESect-7.8 P-32ESect-7.8 P-33ESect-7.8 P-34ESect-7.8 P-35ESect-7.8 P-36ESect-7.8 P-37ESect-7.8 P-38ESect-7.8 P-39ESect-7.8 P-40ESect-7.8 P-41ESect-7.8 P-42ESect-7.8 P-43ESect-7.8 P-44ESect-7.8 P-45ESect-7.8 P-46ESect-7.8 P-47ESect-7.8 P-48ESect-7.8 P-49ESect-7.8 P-50ESect-7.8 P-51ESect-7.8 P-52ESect-7.8 P-53ESect-7.8 P-54ESect-7.8 P-55ESect-7.8 P-56ESect-7.8 P-57ESect-7.8 P-58ESect-7.8 P-59ESect-7.8 P-60ESect-7.9 P-1CPSect-7.9 P-2CPSect-7.9 P-3CPSect-7.9 P-4CPSect-7.9 P-5CPSect-7.9 P-1SWUSect-7.9 P-2SWUSect-7.9 P-3SWUSect-7.9 P-4SWUSect-7.9 P-5SWUSect-7.9 P-6SWUSect-7.9 P-7SWUSect-7.9 P-8SWUSect-7.9 P-9SWUSect-7.9 P-10SWUSect-7.9 P-1ESect-7.9 P-2ESect-7.9 P-3ESect-7.9 P-4ESect-7.9 P-5ESect-7.9 P-6ESect-7.9 P-7ESect-7.9 P-8ESect-7.9 P-9ESect-7.9 P-10ESect-7.9 P-11ESect-7.9 P-12ESect-7.9 P-13ESect-7.9 P-14ESect-7.9 P-15ESect-7.9 P-16ESect-7.9 P-17ESect-7.9 P-18ESect-7.9 P-19ESect-7.9 P-20ESect-7.9 P-21ESect-7.9 P-22ESect-7.9 P-23ESect-7.9 P-24ESect-7.9 P-25ESect-7.9 P-26ESect-7.9 P-27ESect-7.9 P-28ESect-7.9 P-29ESect-7.9 P-30ESect-7.9 P-31ESect-7.9 P-32ESect-7.9 P-33ESect-7.9 P-34ESect-7.9 P-35ESect-7.9 P-36ECh-7 P-1RECh-7 P-2RECh-7 P-3RECh-7 P-4RECh-7 P-5RECh-7 P-6RECh-7 P-7RECh-7 P-8RECh-7 P-9RECh-7 P-10RECh-7 P-11RECh-7 P-12RECh-7 P-13RECh-7 P-14RECh-7 P-15RECh-7 P-16RECh-7 P-17RECh-7 P-18RECh-7 P-19RECh-7 P-20RECh-7 P-21RECh-7 P-22RECh-7 P-23RECh-7 P-24RECh-7 P-25RECh-7 P-26RECh-7 P-27RECh-7 P-28RECh-7 P-29RECh-7 P-30RECh-7 P-31RECh-7 P-32RECh-7 P-33RECh-7 P-34RECh-7 P-35RECh-7 P-36RECh-7 P-37RECh-7 P-38RECh-7 P-39RECh-7 P-40RECh-7 P-41RECh-7 P-42RECh-7 P-43RECh-7 P-44RECh-7 P-45RECh-7 P-46RECh-7 P-47RECh-7 P-48RECh-7 P-49RECh-7 P-50RECh-7 P-51RECh-7 P-52RECh-7 P-53RECh-7 P-54RECh-7 P-55RECh-7 P-56RECh-7 P-57RECh-7 P-58RECh-7 P-59RECh-7 P-60RECh-7 P-61RECh-7 P-62RECh-7 P-63RECh-7 P-64RECh-7 P-65RECh-7 P-66RECh-7 P-67RECh-7 P-68RECh-7 P-69RECh-7 P-70RECh-7 P-71RECh-7 P-72RECh-7 P-73RECh-7 P-74RECh-7 P-75RECh-7 P-76RECh-7 P-77RECh-7 P-78RECh-7 P-79RECh-7 P-80RECh-7 P-81RECh-7 P-82RECh-7 P-83RECh-7 P-84RECh-7 P-85RECh-7 P-86RECh-7 P-87RECh-7 P-88RECh-7 P-89RECh-7 P-90RECh-7 P-91RECh-7 P-92RECh-7 P-93RECh-7 P-94RECh-7 P-95RECh-7 P-96RECh-7 P-97RECh-7 P-98RECh-7 P-99RECh-7 P-100RECh-7 P-101RECh-7 P-102RECh-7 P-103RECh-7 P-104RECh-7 P-1TYSCh-7 P-2TYSCh-7 P-3TYSCh-7 P-4TYSCh-7 P-5TYSCh-7 P-6TYSCh-7 P-7TYSCh-7 P-8TYSCh-7 P-9TYSCh-7 P-10TYSCh-7 P-11TYSCh-7 P-12TYSCh-7 P-13TYSCh-7 P-14TYSCh-7 P-15TYSCh-7 P-16TYSCh-7 P-17TYSCh-7 P-18TYSCh-7 P-19TYSCh-7 P-20TYSCh-7 P-21TYSCh-7 P-22TYS

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