[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"detail-sidebar-cat-0-en-105":3,"doc-seo-207513-105":59,"doc-detail-207513-en":124},{"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":117,"head_meta":119,"extra_data":121,"updated_unix":123},105,"en","gcse-further-maths-optimising-problems","GCSE Further Maths - Optimising Problems","","Revision sheet for GCSE Further Maths on optimisation problems, guiding learners to read questions carefully, check results, and show full working. Covers maximising and minimising areas and volumes for rectangles, composite shapes made from rectangles, and cuboids, using perimeter and surface area constraints. Includes worked prompt steps to form algebraic expressions, differentiate, find critical values, and confirm whether each optimum is a maximum or minimum, plus an application to an open-topped tank.",{"@graph":69,"@context":116},[70,84,99],{"@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/gcse-further-maths-optimising-problems/207513/",{"url":83,"name":65,"@type":85,"author":86,"headline":65,"publisher":89,"fileFormat":92,"inLanguage":63,"description":67,"dateModified":93,"datePublished":93,"encodingFormat":92,"isAccessibleForFree":94,"interactionStatistic":95},"DigitalDocument",{"name":87,"@type":88},"Asher","Person",{"url":74,"name":90,"@type":91},"DocShare","Organization","application/pdf","2026-09-05",true,{"@type":96,"interactionType":97,"userInteractionCount":4},"InteractionCounter",{"@type":98},"ViewAction",{"@type":100,"mainEntity":101},"FAQPage",[102,108,112],{"name":103,"@type":104,"acceptedAnswer":105},"What do you need to do for each optimisation question?","Question",{"text":106,"@type":107},"Read the question carefully, check your answer makes sense, and always show your working throughout the solution.","Answer",{"name":109,"@type":104,"acceptedAnswer":110},"How are maximum and minimum values identified in these problems?",{"text":111,"@type":107},"Form an expression for the quantity to optimise (area or volume), then determine critical values and verify whether the result is a maximum or minimum.",{"name":113,"@type":104,"acceptedAnswer":114},"What geometric solids appear in the optimisation questions?",{"text":115,"@type":107},"Rectangles and composite shapes made from two rectangles, plus cuboids, including a metal box and an open-topped tank.","https://schema.org",{"og:url":83,"og:type":118,"og:title":65,"og:site_name":90,"og:description":67},"article",{"robots":120,"canonical":83},"index,follow",{"doc_id":122,"site_id":62},207513,1788599813,{"code":4,"msg":5,"data":125},{"doc_id":122,"user_id":126,"nickname":87,"user_avatar":127,"doc_module":4,"category_id":19,"category_name":20,"doc_title":65,"doc_description":67,"doc_content":128,"file_id":129,"file_url":130,"file_type":131,"file_size":132,"view_count":4,"is_deleted":4,"is_public":8,"is_downloadable":8,"audit_status":8,"page_count":39,"language":133,"language_code":63,"site_id":62,"html_lang":63,"table_of_contents":134,"faqs":135,"seo_title":136,"seo_description":67,"update_tm":123,"read_time":46},687197207639,"https://ap-avatar.wpscdn.com/davatar_a8503ba1806abce46bf441b54a3ca4cd","| Name: |  |\n| --- | --- |\n| GCSE Further Maths\u003Cbr>Optimising Problems | |\n| Ensure you have: Pencil, Pen, Calculator |  |\n\nGuidance  \n1. Read each question carefully before you begin answering it.  \n2. Check your answers seem right.  \n3. Always show your workings  \nRevision for this topic  \n[www.corbettmaths.com/gcse-further-maths](www.corbettmaths.com/gcse-further-maths)  \n© Corbettmaths 2024  \n1. A farmer creates a pen for his chickens.  \nThe width of the field is x metres.  \nThe perimeter of the field is 100 metres.  \n(a) Show that the length of the rectangle is  \n50 − x  \nmetres  \n(1)  \n(b) Show that the area of the field is A = 50x − x 2  \n(1)  \n(c) Find the value of x for which A is a maximum and show it is a maximum.  \n(5)  \n© Corbettmaths 2024  \n2. The shape below is made from two rectangle.  \nThe perimeter of the shape is 100cm.  \n(a) Show that y = 45 − 5x  \n(2)  \nThe area of the shape is Acm2  \n(b) Show that A = 225 + 116x − 13x2  \n(2)  \n(c) Find the value of x for which A is a maximum and show it is a maximum.  \n(5)  \n© Corbettmaths 2024  \n3. Shown below is a metal box in the shape of a cuboid.  \nThe volume of the box is 80cm³  \n(a) Show that y = ~~ ~~8x02  \n(2)  \n(b) Show that the area of metal to make the box is given by  \nA = 2x2 +  320  \nx  \n(2)  \n(c) Find the value of x for which A is a minimum, and show it is a minimum.  \n(6)  \n© Corbettmaths 2024  \n4. Shown below is a cuboid  \nThe surface area of the cuboid is 120cm² .  \n(a) Show that y = 20x − ~~ ~~2x3  \n(3)  \n(b) Show that the volume of the cuboid is given by V = 40x − ~~ ~~43~~ ~~ x 3  \n(2)  \n(c) Find the value of x for which V is a maximum, and show it is maximum.  \n(5)  \n© Corbettmaths 2024  \n(d) Use your answer to (c) to find the maximum volume of the cuboid  \n  cm²  \n(2)  \n5. The volume of a container with a height of x, is given by V = x(x − 1)(9 − x) where 1 \u003C x \u003C 9  \n(a) Find ddVx  \n(3)  \n(b) Hence find the value of x for which the volume is a maximum.  \nGive your answer to 1 decimal place.  \n(3)  \n© Corbettmaths 2024  \n6. An open-topped tank in the shape of a cuboid is shown below.  \nThe surface area of the cuboid is 300cm²  \n(a) Show that y = ~~ ~~50x − ~~ ~~x3  \n(3)  \n(b) Show that the volume of the tank is V = 100x − ~~ ~~23~~ ~~ x 3  \n(3)  \n(c) Find the value of x for which V is a maximum  \n(3)  \n© Corbettmaths 2024  \n(d) Show the answer is (c) is a maximum.  \n(2)  \n(e) Find the maximum volume of the tank  \n(2)  \n© Corbettmaths 2024","cbCaivo7Ac5jYSz4","https://ap.wps.com/l/cbCaivo7Ac5jYSz4","pdf",309559,"English","# Revision for this topic\n# Optimising Problems\n## Question 1: Rectangle field\n## Question 2: Composite rectangles shape\n## Question 3: Cuboid metal box\n## Question 4: Cuboid with surface area\n## Question 5: Container volume optimisation\n## Question 6: Open-topped cuboid tank","[{\"question\":\"What do you need to do for each optimisation question?\",\"answer\":\"Read the question carefully, check your answer makes sense, and always show your working throughout the solution.\"},{\"question\":\"How are maximum and minimum values identified in these problems?\",\"answer\":\"Form an expression for the quantity to optimise (area or volume), then determine critical values and verify whether the result is a maximum or minimum.\"},{\"question\":\"What geometric solids appear in the optimisation questions?\",\"answer\":\"Rectangles and composite shapes made from two rectangles, plus cuboids, including a metal box and an open-topped tank.\"}]","GCSE Further Maths - Optimising Problems | PDF"]