[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"detail-sidebar-cat-0-en-105":3,"doc-seo-522872-105":59,"doc-detail-522872-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","optimization-infinite-calculus-solve-each-optimization-problem","Optimization - Infinite Calculus - Solve each optimization problem","","Optimization worksheet from Kuta Software-Infinite Calculus with three word problems. It instructs solving each problem by using a sketch box for setup and a graph for a single-variable function to be minimized or maximized. Problems include maximizing the volume of an open-top box built from a 14 in by 30 in cardboard sheet, finding the point on y = x closest to (7, 0), and minimizing the total rope length needed to anchor two poles of heights 16 ft and 24 ft with a common stake.",{"@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/optimization-infinite-calculus-solve-each-optimization-problem/522872/",{"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/optimization-infinite-calculus-solve-each-optimization-problem/522872.png","ImageObject",300,407,{"name":92,"@type":93},"Angel","Person",{"url":74,"name":95,"@type":96},"DocShare","Organization","application/pdf","2026-10-22","2026-10-08",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},"What method does the worksheet suggest for solving the optimization problems?","Question",{"text":112,"@type":113},"It suggests sketching the setup with the provided box and sketching the one-variable function to be minimized or maximized using the provided graph.","Answer",{"name":115,"@type":110,"acceptedAnswer":116},"How is the volume of the box modeled in the first problem?",{"text":117,"@type":113},"The worksheet defines the volume as V = (30 − 2x)(14 − 2x)·x, with 0 \u003C x \u003C 7, where x is the square side length cut from each corner.",{"name":119,"@type":110,"acceptedAnswer":120},"Where should the stake be placed in the rope-length optimization problem?",{"text":121,"@type":113},"The stake should be placed 12 ft from the short 16 ft pole (or 18 ft from the long 24 ft pole) to use the least amount of rope.","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},522872,1792299495,{"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":29,"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,"preview_page_count":4},687207412472,"https://ap-avatar.wpscdn.com/davatar_155a257f0dc6eb9ab79c44ca47cae57d","Kuta Software-Infinite Calculus Name   \nOptimization Date  Period   \nSolve each optimization problem. You may use the provided box to sketch the problem setup and the provided graph to sketch the function of one variable to be minimized or maximized.  \n1) A supermarket employee wants to construct an open-top box from a 14 by 30 in piece of cardboard. To do this, the employee plans to cut out squares of equal size from the four corners so the four sides can be bent upwards. What size should the squares be in order to create a box with the largest possible volume?  \n2) Which point on the graph of y = x is closest to the point (7, 0) ?  \n3) Two vertical poles, one 16 ft high and the other 24 ft high, stand 30 feet apart on a flat field. A worker wants to support both poles by running rope from the ground to the top of each post. If the worker wants to stake both ropes in the ground at the same point, where should the stake be placed to use the least amount of rope?  \nKuta Software-Infinite Calculus Name   \nOptimization Date  Period   \nSolve each optimization problem. You may use the provided box to sketch the problem setup and the provided graph to sketch the function of one variable to be minimized or maximized.  \n1) A supermarket employee wants to construct an open-top box from a 14 by 30 in piece of cardboard. To do this, the employee plans to cut out squares of equal size from the four corners so the four sides can be bent upwards. What size should the squares be in order to create a box with the largest possible volume?  \n\n| 14 in |  |\n| --- | --- |\n| 30 in |  |\n\n1 2 3  \nV = the volume of the box x = the length of the sides of the squares Function to maximize: V = (30 − 2x)(14 − 2x) ⋅ x where 0 \u003C x \u003C 7  \nSides of the squares: 3 in  \n4  \n5  \n6  \n7  \nx  \n2) Which point on the graph of y = x is closest to the point (7, 0) ?  \n14  \n13  \n12  \n11  \n10  \n9  \n8  \n7  \n6  \n5  \n4  \n3  \n2  \n1  \n\n|  y\u003Cbr>| |\n| --- | --- |\n|  |  |\n\n1 2 3 4 5 6 7 8 9 10 11 12 13 14 x 1 2 3 4 5 6 7 8 9 10 11 12 13 14 xd = the distance from point (7, 0) to a point on the curve x = the x-coordinate of a point on the curve Function to minimize: d = (x − 7)2 + ( x )2 where − ∞ \u003C x \u003C ∞  \nPoint on the curve that is closest to the point (7, 0) : (~~ ~~132~~ ~~ , ~~ ~~262)  \n3) Two vertical poles, one 16 ft high and the other 24 ft high, stand 30 feet apart on a flat field. A worker wants to support both poles by running rope from the ground to the top of each post. If the worker wants to stake both ropes in the ground at the same point, where should the stake be placed to use the least amount of rope?  \n\n| |\n| --- |\n| 30 ft\u003Cbr>|\n\n150  \n140  \n130  \n120  \n110  \n100  \n90  \n80  \n70  \n60  \n50  \n40  \n30  \n20  \n10  \n\n|  L ( if x = the horizontal distance from the short pole to the stake |  |  |  |  |  |  |  |  |  |  |  |  |  |  | ) |\n| --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- |\n|  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |\n|  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |\n|  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |\n|  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |\n|  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |\n|  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |\n|  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |\n|  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |\n|  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |\n| |  | |  |  |  |  |  |  |  |  | | | | |  |\n|  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |\n|  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |\n|  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |\n|  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |\n|  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |\n\n2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 x  \nL = the total length of rope x = the horizontal distance from the short pole to the stake Function to minimize: L = x 2 + 162 + (30 − x)2 + 242 where 0 ≤ x ≤ 30  \nStake should be placed: 12 ft from the short pole (or 18 ft from the long pole)  \nCreate your own worksheets","cbCaicHacsUkVZzr","https://ap.wps.com/l/cbCaicHacsUkVZzr","pdf",48122,"English","# Open-top box volume maximization\n# Closest point on y = x\n# Minimize total rope length for two poles","[{\"question\":\"What method does the worksheet suggest for solving the optimization problems?\",\"answer\":\"It suggests sketching the setup with the provided box and sketching the one-variable function to be minimized or maximized using the provided graph.\"},{\"question\":\"How is the volume of the box modeled in the first problem?\",\"answer\":\"The worksheet defines the volume as V = (30 − 2x)(14 − 2x)·x, with 0 \\u003c x \\u003c 7, where x is the square side length cut from each corner.\"},{\"question\":\"Where should the stake be placed in the rope-length optimization problem?\",\"answer\":\"The stake should be placed 12 ft from the short 16 ft pole (or 18 ft from the long 24 ft pole) to use the least amount of rope.\"}]","Optimization - Infinite Calculus - Solve each optimization problem | PDF",1791426258,15]