MTH 124 Focus Session 1 Modeling Economics with Functions Goals cid 136 Use linear functions to model economics cid 136 Understand the concept of cost revenue and pro t cid 136 Get familiar with Desmos Pre activity Do before class Cost Revenue and Pro t are fundamental concepts in business economics Familiarize yourself with their de nitions below De nitions The Cost function denoted C q gives the total cost of producing a quantity q of some item It can be broken into sub costs T otal Cost F ixed Costs V ariable Costs cid 136 Fixed costs Costs that do not depend on the quantity of items produced Examples of xed costs include rent machinery costs permits etc cid 136 Variable costs Costs that do depend on the quantity of items produced Examples include the labor costs raw materials and shipping costs The Revenue function R q gives the total amount of money earned by a company from selling q items Total revenue is the price of each item times the number of items sold Revenue P rice Quantity The Pro t function H q gives the total pro t from producing and selling q items Pro t is total revenue minus total cost H q R q C q cid 136 Question if both the cost function and the revenue function are linear functions is the pro t function also a linear function Why cid 136 Get started with Desmos Go to the website https www desmos com calculator Plot the graphs of the functions f x 3x 5 and g x 9 2x Find the x value where the two graphs intersect How could you nd this value without using the graphs 1 MTH 124 Focus Session 1 Modeling Economics with Functions In Class Activity Consider a small company SnowMow that produces snowboards In January the expenditures at SnowMow can be broken down as 1600 for rent 1000 for advertising and for each snowboard produced they spend 250 in supplies and labor 1 What are the xed costs of SnowMow in January What are the variable costs of SnowMow in January Hint The variable cost should depend on q the number of snowboards produced 2 Find a linear equation that models SnowMow s total cost C q in dollars of producing q snowboards in January 3 The owner has 27600 capital to invest in SnowMow in January What is the maximum number of snowboards that SnowMow can produce in January assuming the owner invests all of the available capital into the business 40000 30000 20000 10000 s r a l l o d n i t s o c l a t o T 4 Use Desmos to plot the graph of C q Transfer the graph onto the axes to the right Pay close attention to the units 0 20 40 60 80 100 120 Number of snowboards q 2 MTH 124 Focus Session 1 Modeling Economics with Functions 5 Find the y intercept for the January cost function C q above Interpret what this number means in terms of cost Continuing with our example suppose SnowMow sells their snowboards for 350 each 6 Find an equation for SnowMow s revenue function R q in the month of January in terms of q the number of snowboards sold The quantity of goods at which the total cost equals the total revenue is called the break even point At that quantity the business is neither making money nor losing money 7 Solve C q R q for the break even point Plot both C q and R q in Desmos Label the break even point 8 Find an equation for SnowMow s pro t function H q in January 9 Find the x intercept of H q using Desmos Compare it with the break even point in Question 7 3