In the Herb Business, Part I

63 In the Herb Business, Part I A. You have joined a highl respected St Croi herbalist in a business to market her herbal products. Your personal goal is to assure that the business thrives. Researchers from the US and elsewhere come to the Caribbean to stud traditional use of herbs; ou want to make sure that people here get full benefit of the traditional knowledge. One item ou are selling is an herbal medicine based on simperviv known elsewhere as aloe vera). Simperviv is a native Virgin Islands plant that, when properl prepared b a skilled herbalist, can be used as a purgative, as treatment for stomach ills, and as an especiall effective eternal medicine for burns and cuts. It is also valuable as a hair conditioner. On average, ou sell 0 bottles per week of the herbalist s simperviv decoction. You have observed that when ou raise the price of this ecellent medicine, ou sell less of it. For each dollar per bottle ou raise our price, ou sell about 5 fewer bottles per week. Let be the number of dollars ou increase our price and let represent the number of bottles ou sell per week, on average.. Complete the following table. 0 3 4 0 0 5 0. Plot the points from the table on the graph below and connect them with a straight line. Before plotting the points, choose appropriate scales to fit this grid and suit the data in our table. Label the aes. 3. Develop an equation for in terms of. 4. the -intercept and how does it relate to the contet of the problem? Rosalie A. Dance and James T. Sandefur, 004

64 5. the slope and how does it relate to the contet of the problem? 6. How man bottles would ou epect to sell if ou reduced our price b $? 7. How man bottles would ou epect to sell if ou raised our price b 40 cents? 8. Draw a rise/run triangle between the points, ) =, 5) and, ) =5, 95)., what does it represent in this contet, and where is it represented on the triangle? and what does it represent? and what does it represent, giving units? Wh is that result epected? 9. Repeat problem 8 using, ) =, 0) and, ) =0, 70). B. You also sell noni juice b the glass in our herb shop. Now, ou sell about 90 glasses per week. You notice that for each 0 cents 0. dollar) ou raise the price per glass, ou sell fewer glasses per week.. How man glasses of noni juice will ou sell if ou raise the price b dollars?. Draw a line on the following graph where represents the number of dollars ou increase the price and represents the number of glasses ou sell. Plotting a few points first will help. Make sure ou pick an appropriate scale for the aes. Label them. 3. Draw a rise/run triangle between the points, ) =0, 90) and, ) =0.5, 80). What is, what does it represent in this contet, and where is it represented on the triangle? glasses Rosalie A. Dance and James T. Sandefur, 004

65 and what does it represent? and what does it represent, giving units? Is this result epected? 4. Give an equation for this line. the -intercept and how does it relate to the contet? the slope and how does it relate to the contet. Rosalie A. Dance and James T. Sandefur, 004

66 Teaching Guide for In the Herb Business, Part I Introduction: The mathematical content of this lesson focuses on slopes and -intercepts of linear functions. The student will make a table of values, a graph, and an equation to represent a linear situation. The will analze the meaning of the slope in the contet of the meaning of the variables. Similarl, the will observe the -intercept and interpret its meaning in the contet of the situation. A..On average, ou sell 0 bottles per week of the herbalist s simperviv decoction. You have observed that when ou raise the price of this ecellent medicine, ou sell less of it. For each dollar per bottle ou raise our price, ou sell about 5 fewer bottles per week. Let be the number of dollars ou increase our price and let represent the number of bottles ou sell per week, on average.. Complete the following table. 0 3 4 0 0 5 0 05 00 70. Plot the points from the table on the graph below and connect them with a straight line. Before plotting the points, choose appropriate scales to fit this grid and suit the data in our table. Label the aes. The scale is often a challenge for students. Here is an eample. 00 80 60 40 0 4 6 8 0 3. Develop an equation for in terms of. = 0 5. Giving students time to work this out for themselves will facilitate their abilit to interpret the -intercept and the slope in the following questions. 4. the -intercept and how does it relate to the contet of the problem? The line intercepts the -ais at 0, 0). In the contet of the problem, this means that if ou don t change the price if = 0), ou will sell 0 bottles of the herbal product. Rosalie A. Dance and James T. Sandefur, 004

67 5. the slope and how does it relate to the contet of the problem? The slope is -5. This indicates that ever time ou increase the value of b, decreases b 5; in contet, ever time ou increase the price b $, the average number of bottles sold in a month decreases b 5. 6. How man bottles would ou epect to sell if ou reduced our price b $? 5, since = 0 5-) = 5. 7. How man bottles would ou epect to sell if ou raised our price b 40 cents? Since 40 cents is $0.4, we evaluate = 0 50.4) = 8 and epect to sell 8 bottles. 8. Draw a rise/run triangle between the points, ) =, 5) and, ) =5, 95). The visual helps man students with the interpretation., what does it represent in this contet, and where is it represented on the triangle? = 95 5= 0 bottles. In contet, it represents a decrease in the number of bottles ou ll sell if ou increase the price b $5 from the original price after having alread increased it b $. On the graph, it is illustrated b the length of the vertical side of the triangle, the difference in height between the points. Since the second point is below the first, the path along the vertical side as ou traverse from the first point to the second is downward. and what does it represent? = 5 = 4 dollars. In contet, it represents the difference between a $5 increase and a $ increase in price. On the graph, it is illustrated b the horizontal side of the triangle, the horizontal distance between the points. and what does it represent, giving units? Wh is that result epected? Using the results we just found, 0 bottles = = 5 4 dollars bottles dollar. We epected that result because we knew that an increase in price/bottle of $ would decrease the number of sales b 5. 9. Repeat problem 8 using, ) =, 0) and, ) =0, 70). = 70 0= 40 bottles. This represents a decrease in the number of bottles ou ll sell if ou increase the price b $0 from the original price following an increase b $. On the graph, it is illustrated b the length of the vertical side of the triangle and the downward direction of the change as ou go from the first point to the second. = 0 = 8 dollars. In contet, it represents the difference between a $0 increase and a $ increase in price. On the graph, it is illustrated b the length of the horizontal side of the triangle. Rosalie A. Dance and James T. Sandefur, 004

68 Using the results we just found, 40 bottles = = 5 8 dollars bottles dollar. We epected that result because we knew that an increase in price/bottle of $ would decrease the number of sales b 5, and we are happ to see that it holds with the new numbers!. B. You also sell noni juice b the glass in our herb shop. Now, ou sell about 90 glasses per week. You notice that for each 0 cents 0. dollar) ou raise the price per glass, ou sell fewer glasses per week.. How man glasses of noni juice will ou sell if ou raise the price b dollars? Since $ = 0 0 cents) and ou sell fewer glasses for ever 0 cent increase in price, ou epect to sell 0 = 40 fewer glasses. You ll then sell 90 40 = 50 glasses,. Draw a line on the following graph where represents the number of dollars ou increase the price and represents the number of glasses ou sell. Plotting a few points first will help. Make sure ou pick an appropriate scale for the aes. Label them. 00 50 glasses -.40.0.40.60.80 3. Draw a rise/run triangle between the points, ) =0, 90) and, ) =0.5, 80)., what does it represent in this contet, and where is it represented on the triangle? =80 90 = -0 glasses. This represents a decrease of 0 glasses of noni juice each week. It is represented in the vertical side of the rise/run triangle. and what does it represent? =0.5 0 =.5 dollars. This represents an increase of 50 cents in the cost of a glass of noni juice. Rosalie A. Dance and James T. Sandefur, 004

69 and what does it represent, giving units? Is this result epected? 80-90 0 glasses = = = 0 glasses/dollar, that is, a decrease in sales of 0.5 0.5 dollars glasses for each dollar increase in price. 4. Give an equation for this line. the -intercept and how does it relate to the contet? the slope and how does it relate to the contet. = 90 0. The -intercept is 0, 90); it tells us that if there is no change in price, ou sell 90 glasses of noni juice a week on average). The slope is -0; it tells us that ou will decrease sales b 0 glasses if ou increase the cost of a glass of juice b $. Rosalie A. Dance and James T. Sandefur, 004