Lesson Linear Models Answering Various Types of Questions Algebraically

Transcription

1 STTWY STUDENT HNDOUT STUDENT NME DTE INTRODUCTION Many people make small items at home to sell at craft sales or on ebay or through web pages. In some cases, their costs and revenues are easily determined. In this example, Jenna makes scented oil lamps. Her fixed costs for her lamp-making business are $1100 per month for rent and utilities. The materials needed to make one scented oil lamp costs $4.50. She sells each lamp for $ ssume that she can sell all the lamps in one month that she can make. Our main question here is how many lamps should she make in order to make a profit, which means that her revenue exceeds her costs? long the way, we will answer some other questions. Let x = the number of lamps produced in one month. Let C = Jenna s total cost for her lamp-making business for one month, when she makes x lamps. Let R = Jenna s revenue for her lamp-making business for one month, when she sells x lamps. We will find formulas for the Cost and Revenue, which, for this situation are linear models. While most of the questions we might be interested in here could be answered from a table or graph of these two formulas, we will also use the formulas to algebraically answer these questions. That is faster than making tables and graphs to estimate answers such questions and also gives more precise answers. Solution: We will find the cost formula in the following way: 1 Suppose she has the space already rented and set up to make lamps, but in this particular month, she does not actually buy the materials for any lamps and does not make any lamps. What is her cost? What is her revenue this month?

2 STTWY STUDENT HNDOUT 2

3 STTWY STUDENT HNDOUT 3 2 Suppose she makes and sells exactly one lamp in a particular month. What is her cost? What is her revenue this month? Continuing this idea, we fill out the following table: x C R *4.50 2* etc. x x*4.50 x*10.50 Thus, we find these two linear formulas: and. TRY THESE Use the formulas for cost and revenue to write an algebraic equation to solve and then solve it. Then check your solution to see if it really does work. 1 If she makes and sells 200 lamps in one month: What is her cost? B What is her revenue? 2 If her revenue one month is $2016, how many lamps are sold?

4 STTWY STUDENT HNDOUT 4 3 If her cost one month is $2090, how many lamps are made? 4 How many lamps must she make and sell to break even? (Hint: That means that she takes in exactly as much money as she has paid out in costs.) Inequalities Considering Jenna and her lamps: 5 Write an algebraic inequality which, if you solve it, would answer the question: For what numbers of lamps will she make money?

5 STTWY STUDENT HNDOUT 5 TKE IT HOME 1 Peter has a paper route. He buys the papers he delivers for $0.32 each, and charges his customers $0.75 each. His fixed costs per day (gas for the car and an appropriate share of maintenance on the car) are $9. Because he is delivering these papers, he knows how many to buy each day, so assume that he sells each paper he buys. Let x = the number of papers he buys and sells each day. How many papers will he need to buy/sell per day in order to break even? Explain your answer. B If he would like to consider increasing his costs include paying himself $10/hour for 2 hours to deliver the papers, then how many papers would he need to buy/sell to break even? (*Hint: nswer this first: How much will his cost increase?)

6 STTWY STUDENT HNDOUT 6 2 The law of supply and demand says that, in a free economy, if a commodity is popular, that is, there is a lot of demand for it, then that will drive the prices up, so more companies will go into business supplying that product, which increases the supply. When the supply is increased, then there is more supply than demand and that causes the price to drop. When the prices drop, then some of the suppliers are no longer interested in pursuing that business and so the supply decreases. In these situations, there is often a price at which the market is in equilibrium, which is when the supply and demand are equal. For farm commodities, the laws of supply and demand are often used to explain the interactions of farmers and consumers. Let x = the number of million bushels of millet and y = the current market price of a bushel of millet. The supply of millet is modeled by.. The demand for millet is modeled by If the market price is $7.80, how many bushels will be supplied? B If the market price is $7.80, what will the demand be? C How many bushels are being supplied when the market reaches equilibrium? D Find the market price at which this market reaches equilibrium.

7 STTWY STUDENT HNDOUT This lesson is part of STTWY, Pathway Through College Statistics, which is a product of a Carnegie Networked Improvement Community that seeks to advance student success. Version 1.0, Pathway Through Statistics, Statway was created by the Charles. Dana Center at the University of Texas at ustin under sponsorship of the Carnegie Foundation for the dvancement of Teaching. This version 1.5 and all subsequent versions, result from the continuous improvement efforts of the Carnegie Networked Improvement Community. The network brings together community college faculty and staff, designers, researchers and developers. It is an open-resource research and development community that seeks to harvest the wisdom of its diverse participants in systematic and disciplined inquiries to improve developmental mathematics instruction. For more information on the Statway Networked Improvement Community, please visit carnegiefoundation.org. For the most recent version of instructional materials, visit Statway.org/kernel STTWY and the Carnegie Foundation logo are trademarks of the Carnegie Foundation for the dvancement of Teaching. Pathway Through College Statistics may be used as provided in the CC BY license, but neither the Statway trademark nor the Carnegie Foundation logo may be used without the prior written consent of the Carnegie Foundation.