[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"detail-sidebar-cat-0-en-105":3,"doc-seo-297543-105":59,"doc-detail-297543-en":130},{"code":4,"msg":5,"data":6},0,"success",[7,13,18,23,28,33,38,43,48,51,55],{"id":8,"doc_module":4,"doc_module_name":9,"category_name":10,"show_sort_weight":11,"slug":12},1,"Document","Story & Novel",90,"story-novel",{"id":14,"doc_module":4,"doc_module_name":9,"category_name":15,"show_sort_weight":16,"slug":17},2,"Literature",80,"literature",{"id":19,"doc_module":4,"doc_module_name":9,"category_name":20,"show_sort_weight":21,"slug":22},4,"Exam",70,"exam",{"id":24,"doc_module":4,"doc_module_name":9,"category_name":25,"show_sort_weight":26,"slug":27},5,"Comic",60,"comic",{"id":29,"doc_module":4,"doc_module_name":9,"category_name":30,"show_sort_weight":31,"slug":32},6,"Technology",50,"technology",{"id":34,"doc_module":4,"doc_module_name":9,"category_name":35,"show_sort_weight":36,"slug":37},7,"Healthcare",40,"healthcare",{"id":39,"doc_module":4,"doc_module_name":9,"category_name":40,"show_sort_weight":41,"slug":42},8,"Research & Report",30,"research-report",{"id":44,"doc_module":4,"doc_module_name":9,"category_name":45,"show_sort_weight":46,"slug":47},9,"Religion & Spirituality",20,"religion-spirituality",{"id":46,"doc_module":4,"doc_module_name":9,"category_name":49,"show_sort_weight":46,"slug":50},"World Cup","world-cup",{"id":52,"doc_module":4,"doc_module_name":9,"category_name":53,"show_sort_weight":52,"slug":54},10,"Lifestyle","lifestyle",{"id":56,"doc_module":4,"doc_module_name":9,"category_name":57,"show_sort_weight":24,"slug":58},19,"General","general",{"code":4,"msg":60,"data":61},"ok",{"site_id":62,"language":63,"slug":64,"title":65,"keywords":66,"description":67,"schema_data":68,"social_meta":123,"head_meta":125,"extra_data":127,"updated_unix":129},105,"en","cylinder-area-student","Cylinder Area - Student","","Learners act as packaging engineers for a beverage company’s new 500 ml energy drink. The activity guides students through building a cylindrical model in Nspire, collecting radius–height data, and deriving a surface-area formula using volume constraints. Students then estimate the radius that minimizes total surface area via a scatter plot, and verify the exact minimum using calculus and Nspire CAS by differentiating and solving s1(r)=0. Finally, students compute the optimized radius and height and discuss cost-based implications through reflection and an extension problem.",{"@graph":69,"@context":122},[70,84,105],{"@type":71,"itemListElement":72},"BreadcrumbList",[73,77,79,82],{"item":74,"name":75,"@type":76,"position":8},"https://docshare.wps.com","Home","ListItem",{"item":78,"name":9,"@type":76,"position":14},"https://docshare.wps.com/document/",{"item":80,"name":20,"@type":76,"position":81},"https://docshare.wps.com/document/exam/",3,{"item":83,"name":65,"@type":76,"position":19},"https://docshare.wps.com/document/cylinder-area-student/297543/",{"url":83,"name":65,"@type":85,"image":86,"author":91,"headline":65,"publisher":94,"fileFormat":97,"inLanguage":63,"description":67,"dateModified":98,"datePublished":99,"encodingFormat":97,"isAccessibleForFree":100,"interactionStatistic":101},"DigitalDocument",{"url":87,"@type":88,"width":89,"height":90},"https://docshare.wps.com/thumbnails/cylinder-area-student/297543.png","ImageObject",300,407,{"name":92,"@type":93},"Ophelia","Person",{"url":74,"name":95,"@type":96},"DocShare","Organization","application/pdf","2026-10-21","2026-09-18",true,{"@type":102,"interactionType":103,"userInteractionCount":34},"InteractionCounter",{"@type":104},"ViewAction",{"@type":106,"mainEntity":107},"FAQPage",[108,114,118],{"name":109,"@type":110,"acceptedAnswer":111},"How do you determine the relationship between radius and height for a 500 ml cylinder?","Question",{"text":112,"@type":113},"Use the cylinder volume formula with volume fixed at 500 ml to express height h as a function of radius r. Then height automatically adjusts in the model when the radius slider changes.","Answer",{"name":115,"@type":110,"acceptedAnswer":116},"What method is used to estimate the radius that minimizes surface area?",{"text":117,"@type":113},"A scatter plot of radius versus surface area is used. Students trace the point that shows the smallest surface area and record the corresponding radius.",{"name":119,"@type":110,"acceptedAnswer":120},"How do calculus and Nspire CAS find the exact minimizing radius?",{"text":121,"@type":113},"Define the surface area function s(r) from the spreadsheet, compute its first derivative s1(r), then solve s1(r)=0 to obtain the exact radius. Convert to a decimal if needed and substitute to compute the minimum surface area.","https://schema.org",{"og:url":83,"og:type":124,"og:title":65,"og:site_name":95,"og:description":67},"article",{"robots":126,"canonical":83},"index,follow",{"doc_id":128,"site_id":62},297543,1790185745,{"code":4,"msg":5,"data":131},{"doc_id":128,"user_id":132,"nickname":92,"user_avatar":133,"doc_module":4,"category_id":19,"category_name":20,"doc_title":65,"doc_description":67,"doc_content":134,"file_id":135,"file_url":136,"file_type":137,"file_size":138,"view_count":34,"is_deleted":4,"is_public":8,"is_downloadable":8,"audit_status":8,"page_count":24,"language":139,"language_code":63,"site_id":62,"html_lang":63,"table_of_contents":140,"faqs":141,"seo_title":142,"seo_description":67,"update_tm":143,"read_time":144},7971461741311,"https://ap-avatar.wpscdn.com/avatar/74000253aff267980c6?x-image-process=image/resize,m_fixed,w_180,h_180&k=1779345379180704826","Materials: Cylinder Area – Student.tns  \nIn this activity you and a partner are packaging engineers for a beverage company. Your task is to determine the dimensions of a cylindrical can for the company’s new 500ml energy drink that minimizes the total surface area. You will collect data from a geometric model, estimate the minimum radius from a scatter plot and use calculus and CAS to determine the actual dimensions.  \nTurn on the Nspire unit.  \nFrom the Home screen choose My Documents and open the file: Cylinder Area – Student.tns  \nGo to page 1.2 and review the instructions for the activity.  \nGo to page 1.3. This construction is a geometric model of a cylinder with volume 500 ml. The slider is used to change the radius. The height of the cylinder will automatically be adjusted to keep the volume 500 ml.  \nMove the cursor over the white dot on the slider, the dot will pulse and the cursor changes to an open hand.  \nMaterials: Cylinder Area – Student.tns  \nTo grasp the white dot to move it, press the NavPad button x and hold it until the hand closes.  \nMove the white dot slowly back and forth until you have all the possible values for the radius. The radius and corresponding height will be displayed. The geometric model will also be updated.  \nHow are the radius and height related?  \n\n| Go to page 1.4. As you were changing the radius a list of the radius and height pairs were collected in this spreadsheet.\u003Cbr>You must determine a formula for the surface area (sa) in terms of only the radius (r) .\u003Cbr>First, you need an equation for height (h) in terms of radius (r) . You know the volume is 500 ml so using the formula for volume of a cylinder, write an equation for h in terms of r. |\n| --- |\n| Next, write the formula for the total surface area of a cylinder and then substitute your previous equation to give an equation for sa in terms of r.\u003Cbr>Record your equation: |\n\nMove the cursor to the cell in column C directly below the label sa. Enter the right side of your equation beginning with = .  \nAfter you press · select All Variable References from the dialog box that appears by using ¤on the NavPad. Press · to select OK.  \nThe surface area for each pair of radius and height you gathered will be calculated and displayed in column C  \nMaterials: Cylinder Area – Student.tns  \nMove to page 1.5 and a scatter plot of radius versus surface area appears.  \nYou need to estimate the radius that minimizes the surface area.  \nPress b 5 1 to select Graph Trace. The coordinates of one of the points will be displayed. By using ¡ and ¢ on the NavPad you can move from point to point.  \nDetermine the point that appears to minimize the surface area. Radius =    \nSurface Area =    \nMove to page 2.1. To be sure you have the actual radius, you will use calculus and the CAS features of Nspire.  \nDefine a function for the surface area (s) in terms of r using the equation you used earlier in the spreadsheet..  \nPress b 1 1 and enter s(r)= followed by your equation from  \nthe spreadsheet The function is stored.  \nTo find the minimum surface area you must find the first derivative.  \nDefine another function s1(r) to be the first derivative. Press b 1 1 and enter s1(r)=  \nPress b 5 1 to insert a derivative expression.  \nMaterials: Cylinder Area – Student.tns  \nIn the denominator type r press e and type s(r) and press ·  \n.  \nThe derivative function is stored.  \nTo calculate the radius that minimizes the surface area you solve the derivative function for zero.  \nPress b 4 1 and type s1(r)=0, r and press ·. The exact radius is displayed.  \nTo see a decimal equivalent press / and then ·.  \nThe dimensions of the cylinder that minimizes surface area are:  \nRadius =   Height =  (Hint: use an earlier equation)  \nTo calculate the minimum surface area you must substitute the radius into the surface area formula.  \nType s(r ׀ The vertical line symbol means \"such that\" and is a grey key found in the top row of the Nspire keyboard.  \nThe value of r is on the screen fr","cbCaik7wMZFS2oEQ","https://ap.wps.com/l/cbCaik7wMZFS2oEQ","pdf",133990,"English","# Radius–height relationship\n## Derive height h from volume\n## Derive surface area sa from r\n# Estimate minimum using scatter plot\n## Trace point with smallest surface area\n# Find exact minimum using calculus and CAS\n## Define s(r) and s1(r)\n## Solve s1(r)=0 for radius\n# Minimum dimensions and surface area\n## Substitute to compute minimum sa\n# Reflection and extension\n## Discuss design trade-offs\n## Re-solve with different cost weighting","[{\"question\":\"How do you determine the relationship between radius and height for a 500 ml cylinder?\",\"answer\":\"Use the cylinder volume formula with volume fixed at 500 ml to express height h as a function of radius r. Then height automatically adjusts in the model when the radius slider changes.\"},{\"question\":\"What method is used to estimate the radius that minimizes surface area?\",\"answer\":\"A scatter plot of radius versus surface area is used. Students trace the point that shows the smallest surface area and record the corresponding radius.\"},{\"question\":\"How do calculus and Nspire CAS find the exact minimizing radius?\",\"answer\":\"Define the surface area function s(r) from the spreadsheet, compute its first derivative s1(r), then solve s1(r)=0 to obtain the exact radius. Convert to a decimal if needed and substitute to compute the minimum surface area.\"}]","Cylinder Area - Student | PDF",1789745620,13]