If you are the manufacturer of the container, you would probably want to use the measurements that give you the least amount of surface area while maintaining the same volume. This would minimize the cost of making the container. This lab is concerned with the minimization process. In other words: Volume is constant. Surface area is changeable. Surface area is a function. Can we determine the minimum value for each cylindrical volume? The table below gives the approximate measurement in inches of some common items that are packed in cylindrical containers. Product Baking power Cleanser Coffee Frosting Pineapple juice Radius (inches) 1.25 1.45 1.95 1.63 2.10 Height (inches) 3,65 7.50 5.20 3.60 6.70 Volume (cubic inches) 17.92 49.54 62.12 30.05 92.82 Part 1 7. Begin with the baking powder. The volume of the baking powder can be expressed as an equation. Find the equation for the volume of a cylinder (or work it out yourself: area of base x height). Set this equation equal to the volume of the baking powder and solve for the heighth with respect to the variable r. & Next, write out the surface area formula. This should contain two independent variables, and h. You can constrain this formula to the given volume by substituting the h with the expression of h(r) you found in step 8. You should now have an equation containing only the variable r. Call this equation S(r) for surface area dependent upon radius. 7 9. Without technology find the derivative of your surface area function (). 10. Graph your derivative in the same viewing window as your surface area function and look at the graph (Set your plot range to show portions that apply to this situation, i.e. no negative radius.) Make sure the window settings on your graph identifies a distinct minimum surface area. Minimum values will occur at x-values that make the derivative equal to 0 or undefined. Mark these coordinates on your graph. 11. Without technology solve for minimized value of r. 12. Use the value for r from step 5 to determine the minimized height and surface area for the baking powder. (Show all work!) Part 2: 13. Repeat the process for the products. Hand in the 5 graphs showing the surface area and derivative of the surface area for each of the products. Be sure to label the product type and identify which graph is which as well as the minimized value. 14. Since you have shown all the work for the Baking Powder container above, you do not need to show all the work for the rest of the containers, just fill in the information in the following table. Product Actual Actual Actual Radius of Height of Minimized Volume Radius Surface Minimized Minimized Surface Area Area area Baking 17.92 1.25 powder area Cleanser 49.54 1.45 Coffee 62.12 1.95 Frosting 30.05 1.63 2.10 Pineapple 92.82 juice 15. When you are done with the calculations, compare the actual surface areas to the minimal surface areas, i.e. calculate the percent difference between the actual SA and the minimized |actual-minimized SA for each product. x 100 actual Enter your results in the following table. If you are the manufacturer of the container, you would probably want to use the measurements that give you the least amount of surface area while maintaining the same volume. This would minimize the cost of making the container. This lab is concerned with the minimization process. In other words: Volume is constant. Surface area is changeable. Surface area is a function. Can we determine the minimum value for each cylindrical volume? The table below gives the approximate measurement in inches of some common items that are packed in cylindrical containers. Product Baking power Cleanser Coffee Frosting Pineapple juice Radius (inches) 1.25 1.45 1.95 1.63 2.10 Height (inches) 3,65 7.50 5.20 3.60 6.70 Volume (cubic inches) 17.92 49.54 62.12 30.05 92.82 Part 1 7. Begin with the baking powder. The volume of the baking powder can be expressed as an equation. Find the equation for the volume of a cylinder (or work it out yourself: area of base x height). Set this equation equal to the volume of the baking powder and solve for the heighth with respect to the variable r. & Next, write out the surface area formula. This should contain two independent variables, and h. You can constrain this formula to the given volume by substituting the h with the expression of h(r) you found in step 8. You should now have an equation containing only the variable r. Call this equation S(r) for surface area dependent upon radius. 7 9. Without technology find the derivative of your surface area function (). 10. Graph your derivative in the same viewing window as your surface area function and look at the graph (Set your plot range to show portions that apply to this situation, i.e. no negative radius.) Make sure the window settings on your graph identifies a distinct minimum surface area. Minimum values will occur at x-values that make the derivative equal to 0 or undefined. Mark these coordinates on your graph. 11. Without technology solve for minimized value of r. 12. Use the value for r from step 5 to determine the minimized height and surface area for the baking powder. (Show all work!) Part 2: 13. Repeat the process for the products. Hand in the 5 graphs showing the surface area and derivative of the surface area for each of the products. Be sure to label the product type and identify which graph is which as well as the minimized value. 14. Since you have shown all the work for the Baking Powder container above, you do not need to show all the work for the rest of the containers, just fill in the information in the following table. Product Actual Actual Actual Radius of Height of Minimized Volume Radius Surface Minimized Minimized Surface Area Area area Baking 17.92 1.25 powder area Cleanser 49.54 1.45 Coffee 62.12 1.95 Frosting 30.05 1.63 2.10 Pineapple 92.82 juice 15. When you are done with the calculations, compare the actual surface areas to the minimal surface areas, i.e. calculate the percent difference between the actual SA and the minimized |actual-minimized SA for each product. x 100 actual Enter your results in the following table