This is the project for learning Optimization Calculus.
I already have some ideas for the first three, but don't know what I should do for the remainings.
If you are also not sure, you can skip some, and please give me a tip for others.
Thanks for your help.
Problem Cans of soft drinks are made from aluminum. Due to the soaring prices, the beverage company executives are looking for all possible cost-cutting measures. You are being asked to find a way to redesign the traditional can to decrease cost as much as possible. The manufacturer fills the containers with 355 cm of liquid and leaves 15 cm3 of open space, so the total volume of 370 cm must be maintained. Part 1: Make a Model of the Can 1. Sketch a right circular cylinder and label the radius, r, and height, h. Give the formulas for volume and surface area of a right circular cylinder. What quantity is known? Take a picture of your work/drawing and include it here. h h = height of can r = radius of can v = volume of the can = 370cm^3 volume is known GIVEN THAT VOLUME OF THE CAN IS 370cm LET, the radius of the cylinder ber Height of the cylinder be h .'. volume = marsh = 370 From this equation we find the height h 370 h = -(1)2. The goal is to minimize the surface area to keep cost as low as possible but note that the formula has too many variables. a. Use the information in [11 to eliminate the height from the surface area formula so that you have surface area as a function of radius only. Show your work. a] SURFACE AREA OF CYLINDER = A = Ei'fl + ETITE 37E] h, = USING \"IE3": IN THE EURFACEAREA OF THE CYLINDER 3TH A = wat + lm-2 = ERIE2} + Efrr2 1T1" 740 I! = [7) + 2m\": b. What is the realistic domain of the surface area function you created in {a}? Domain ofA is Ft-{G} 7" 79 {l Int-(R - [1) Ft = SET OF ALL REAL NUMBER c. Graph the surface area function in m Label each axis by name and units. Include a picture or sketch of your graph. 3. We have been working on finding minimums and maximums of functions over intervals. a. Refer the graph from 2(c) and use desmos to approximate the value of the radius for which the surface area is the least. What is that radius, to the nearest hundredth, and what is the corresponding surface area, to the nearest hundredth? Don't forget units. Radius = Surface Area = b. Now use calculus. Find the derivative of the surface area function and determine the minimum surface area using the derivative. Show your support work. c. Do you get the same radius value as in a and b? Should the values be the same? Explain.4. For the radius which produces the minimum surface area, determine the corresponding dimensions. Round all to the nearest hundredth. Include your work. Radius = Diameter = Height =