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A boutique fruit juice manufacturer produces 2 types of juices, Apple and Fig daily with a total cost function: TC = 6A+AxF+9F where: A is the quantity of the Apple juice (in kegs) and F is the quantity of the Fig juice (in kegs).

A boutique fruit juice manufacturer produces 2 types of juices, Apple and Fig daily with a total cost function: TC = 6A+AxF+9F where: A is the quantity of the Apple juice (in kegs) and F is the quantity of the Fig juice (in kegs).

ENGR.ECONOMIC ANALYSIS
ENGR.ECONOMIC ANALYSIS
14th Edition
ISBN:9780190931919
Author:NEWNAN
Publisher:NEWNAN
Chapter1: Making Economics Decisions
Section: Chapter Questions
Problem 1QTC
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Question 3 Fox Studios is a pest rabbit control company commissioned to assist with a wild rabbit population growth problem on a small island. 1000 rabbits were introduced onto an island, but it was soon realised that the population growth of the rabbits would cause environmental problems on the island. Fox Studios was commissioned to assist with this problem by introducing foxes who help contain the rabbit numbers. The rabbit population growth rate is 10% each month, (This is the net of the natural birth rate and death rate). Each fox eats 10 rabbits per month. (The foxes are also fed by the company as part of their cost). (a) Show that R₁+1 = 1.1 R₁ - 10 F where R, is the rabbit population at month t and F is the number of foxes employed, is a suitable difference equation to use. (b) Verify that R = (1000 - 100 F) × 1.1⁰ + 100 F is the solution to the difference equation. (c) Fox Studios is faced with the following optimisation problem: i. The contract is for 5 years (60 months) ii. Each fox costs $300 000 (special training, transportation, feeding and medical expenses over the 5 years) iii. The company is liable to pay a fine of $1 per rabbit for any excess over 15 000 in every month during the 5 year life. Solve for the number of foxes that should be sent to minimise the cost to the company. Hints: i. ii. iii. There are 11 cases to consider F=0, 1, 2, ... 10 R is an increasing function in t with a calculable time of first breach say T*. An EXCEL spread sheet may be used, but theoretical calculations should be performed.
Transcribed Image Text:Question 3 Fox Studios is a pest rabbit control company commissioned to assist with a wild rabbit population growth problem on a small island. 1000 rabbits were introduced onto an island, but it was soon realised that the population growth of the rabbits would cause environmental problems on the island. Fox Studios was commissioned to assist with this problem by introducing foxes who help contain the rabbit numbers. The rabbit population growth rate is 10% each month, (This is the net of the natural birth rate and death rate). Each fox eats 10 rabbits per month. (The foxes are also fed by the company as part of their cost). (a) Show that R₁+1 = 1.1 R₁ - 10 F where R, is the rabbit population at month t and F is the number of foxes employed, is a suitable difference equation to use. (b) Verify that R = (1000 - 100 F) × 1.1⁰ + 100 F is the solution to the difference equation. (c) Fox Studios is faced with the following optimisation problem: i. The contract is for 5 years (60 months) ii. Each fox costs $300 000 (special training, transportation, feeding and medical expenses over the 5 years) iii. The company is liable to pay a fine of $1 per rabbit for any excess over 15 000 in every month during the 5 year life. Solve for the number of foxes that should be sent to minimise the cost to the company. Hints: i. ii. iii. There are 11 cases to consider F=0, 1, 2, ... 10 R is an increasing function in t with a calculable time of first breach say T*. An EXCEL spread sheet may be used, but theoretical calculations should be performed.
Question 1 A boutique fruit juice manufacturer produces 2 types of juices, Apple and Fig daily with a total cost function: TC = 6A + AxF+9F where: A is the quantity of the Apple juice (in kegs) and F is the quantity of the Fig juice (in kegs). The prices that can be charged are determined by supply and demand forces and are influenced by the quantities of each type of juice according to the following equations: P₁ = 17-A + F for the price (in dollars per keg) of the Apple juice and PF = 23 + 2A-F for the price (in dollars per keg) of the Fig juice. The total revenue is given by the equation: TR = PAX A+ PF XF and the profit given by the equation Profit= TR-TC First, use a substitution of the price variables to express the profit in terms of A and F only. Using the method of Lagrange Multipliers find the maximum profit when total production (quantity) is restricted to 2023 kegs. Note A or F need not be whole numbers. Be sure to show that your solution is a maximum point.
Transcribed Image Text:Question 1 A boutique fruit juice manufacturer produces 2 types of juices, Apple and Fig daily with a total cost function: TC = 6A + AxF+9F where: A is the quantity of the Apple juice (in kegs) and F is the quantity of the Fig juice (in kegs). The prices that can be charged are determined by supply and demand forces and are influenced by the quantities of each type of juice according to the following equations: P₁ = 17-A + F for the price (in dollars per keg) of the Apple juice and PF = 23 + 2A-F for the price (in dollars per keg) of the Fig juice. The total revenue is given by the equation: TR = PAX A+ PF XF and the profit given by the equation Profit= TR-TC First, use a substitution of the price variables to express the profit in terms of A and F only. Using the method of Lagrange Multipliers find the maximum profit when total production (quantity) is restricted to 2023 kegs. Note A or F need not be whole numbers. Be sure to show that your solution is a maximum point.
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