Sec 3.5 Business and Economics Applications. Solution step1 sketch the function. step 2 The primary equation is

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1 Sec 3.5 Business and Economics Applications rev0216 Objectives: 1. Solve business and economics optimization problems 2. Find the price elasticity of demand for demand functions. 3. Recognize basic business terms and formulas - Eample 1: Finding the Maimum Revenue A company has determined that its total revenue (dollars) for a product can be modeled by R where number of units produced (and sold). What production level will yield a maimum revenue Solution step1 sketch the function R y 3e+7 2e+7 1e step 2 The primary equation is R step 3- R is a function on 1 variable No other equation is necessary step 4 -Determine feasible domain When R zero from the equation , Solution is: , positive value Also from the graph appro 550 step 5- maimize the revenue R dr d , 350 units step 6- statement The production level of 350 units will maimize the revenue Definition: Average cost C C 1

2 Eample 2. Finding the minimum Average Cost A company estimates that the cost (dollars) of producing units of product can be modeled by C Find the production level that minimizes the average cost per unit Solution step1 define and sketch (if practical) the function C average cost C C step 2 The primary equation is C step 3- C is already a function on 1 variable no secondary equation required step 4 -Determine feasible domain 0 step 5- mimize the average cost C dc , d , Solution is: 2000, units step 6- statement The production level of 2000 units will minimize the average cost - Do NOW check point 1 Maimize Revenue R Don t need to sketch Given primary equation R 4. Find feasible domain R 0 and solve for where 0 5. Maimize R to find the number of units 6. Statement Asked: What is the maimum revenue? Find revenue for value found in 5. Eample 3 Finding the Maimum Revenue A business sells 2000 units of a product per month at a price of $10 each It can sell 250 more items per month for each $0.25 reduction in price. What price per unit will maimize the monthly revenue? step 1. number of units sold / mo 2

3 p price per unit R monthly revenue step 2 R p primary equation Note: usually the equation for p is given. Here we have to compute it step 3 p $10 means 2000 units p $ units To get the formula for p, use point slope formula change in price p slope m change in p p secondary equation R p primary equation step 4 feasible domain 0 Where is the R 0 ans: 0 and step 5 maimize revenue- find critical number R dr d 6000 units step 6 statement: What were we asked to find? What price per unit will maimize the monthly revenue? p p $6. 0 Production level which maimizes revenue corresponds to a price of $6 per unit Eample 4 Maimum Profit The marketing department of a business has determined the demand for a product can be modeled by p 50 The cost of producting units is given by C What price will yield a maimum profit? step 1 R p, revenue function step 2 P profit R C primary equation 3

4 step 3 R p secondary equation P R C P step4 feasible domain Want possible profit solve radical equation for or graph and estimate the domain step5 maimize P P dp , d Solution is: units. step 6 statement What were we asked? What price will yield a maimum profit? p When is 2500 units, profit is maimized. This corresponds to a selling price of $ 1 per unit. Do NOW # 9 sec Form R p, secondary equation 3. Form primary equation P R C 4. Find feasible domain P 0 and solve for where 0 5. Maimize P to find the number of units 6. Statement What is the price that will maimize profit? Substitute in p and solve for. Look at price elasticity of Demand together. - Price Elasticity of Demand A drop in price might create greater demand for product and thus more Revenue For eample, lowering the price of fresh tomatoes results in selling more tomatoes and is sufficient to increase the revenue. We say the demand is elastic A drop in price might NOT create greater demand and the Revenue might actually decrease. For eample lowering the price of gasoline or coffee might NOT result in greater demand and might NOT be sufficient to increase revenue. We say the demand is inelastic. (not too sensitive to changes in price) 4

5 rate of change of demand Thus price elasticity of demand, rate of change in price is the lower case greek letter, eta Let quantity demanded p price of item rate of change of demand rate of change in price / p/p p/ p/ price elasticity of demand is p/ is lower case greek letter, eta dp/d For a given price, and R revenue function The demand is elastic if 1 dr d 0 R increasing The demand is inelastic if 1 dr d 0 R decreasing Eample 5 Comparing Elasticity and Revenue The demand function for a product is modeled by p 24 2 y a. Find the intervals on which the demand is elastic, inelcastic,and of unit elasticity - p/ dp/d unit elasticity Recall the definition of absolute value c c is equivalent to ( c or c) solving Solution gives

6 solving where the function is defined so chose 144 gives 64, in the domain or This divides the domain into 2 regions 0, 64 and 64, 144 Test a point in each region to determine if 1 or 1 table region 0, 64 64, 144 test value when 0 64 elastic demand 1 when inelastic demand - b. Use the result of part a to describe the behavior of the revenue function R is increasing 1 when 0 64 and R is increasing R is decreasing 1 when and R is decreasing - R is a maimum at 64 units because it changes from increasing to deareasing here R p R R y Do NOW (if time) check point

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