[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"detail-sidebar-cat-0-en-105":3,"doc-seo-522875-105":59,"doc-detail-522875-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","related-rates-related-rates","Related Rates - Related Rates","","Solve a set of six related rates calculus problems involving geometric quantities changing over time. Topics include circular pool area growth from a leaking source, circular oil spill expansion from a tanker rupture, and filling a conical cup with a rising water level. Additional problems model a spherical balloon being inflated and two right-triangle scenarios: a person walking away from a lamppost and an observer tracking a rocket while it rises. Each question asks for an unknown rate at a specified dimension.",{"@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/related-rates-related-rates/522875/",{"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/related-rates-related-rates/522875.png","ImageObject",300,407,{"name":92,"@type":93},"Sarah ","Person",{"url":74,"name":95,"@type":96},"DocShare","Organization","application/pdf","2026-11-11","2026-10-08",true,{"@type":102,"interactionType":103,"userInteractionCount":19},"InteractionCounter",{"@type":104},"ViewAction",{"@type":106,"mainEntity":107},"FAQPage",[108,114,118],{"name":109,"@type":110,"acceptedAnswer":111},"How do you find the rate of change of area when a circle’s radius increases?","Question",{"text":112,"@type":113},"Use A = πr², differentiate with respect to time, and substitute the given dr/dt and the radius value.","Answer",{"name":115,"@type":110,"acceptedAnswer":116},"What equation is used for the conical cup volume when the water height rises?",{"text":117,"@type":113},"Use V = (1/3)πr²h, relate r to h from similar triangles, differentiate with respect to time, then plug in the given dh/dt and the height.",{"name":119,"@type":110,"acceptedAnswer":120},"How are right-triangle related rates problems set up for the shadow and the rocket?",{"text":121,"@type":113},"Define the triangle’s changing legs and hypotenuse, write the Pythagorean relationship (x² + y² = constant forms), differentiate, then solve for the requested rate at the given distance.","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},522875,1792193343,{"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":19,"is_deleted":4,"is_public":8,"is_downloadable":8,"audit_status":8,"page_count":19,"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":52,"preview_page_count":4},962085320529,"https://ap-avatar.wpscdn.com/davatar_9964176cb1d06d4a9deccf72a44ae3dc","Kuta Software-Infinite Calculus Name___________________________________  \nRelated Rates Date________________ Period____  \nSolve each related rate problem.  \n1) Water leaking onto a floor forms a circular pool. The radius of the pool increases at a rate of 4 cm/min. How fast is the area of the pool increasing when the radius is 5 cm?  \n2) Oil spilling from a ruptured tanker spreads in a circle on the surface of the ocean. The area of the spill increases at a rate of 9π m²/min. How fast is the radius of the spill increasing when the radius is 10 m?  \n3) A conical paper cup is 10 cm tall with a radius of 10 cm. The cup is being filled with water so that the water level rises at a rate of 2 cm/sec. At what rate is water being poured into the cup when the water level is 8 cm?  \n4) A spherical balloon is inflated so that its radius (r) increases at a rate of the volume of the balloon increasing when the radius is 4 cm?  \n2 r  \ncm/sec. How fast is  \n5) A 7 ft tall person is walking away from a 20 ft tall lamppost at a rate of 5 ft/sec. Assume the scenario can be modeled with right triangles. At what rate is the length of the person's shadow changing when the person is 16 ft from the lamppost?  \n6) An observer stands 700 ft away from a launch pad to observe a rocket launch. The rocket blasts off and maintains a velocity of 900 ft/sec. Assume the scenario can be modeled as a right triangle. How fast is the observer to rocket distance changing when the rocket is 2400 ft from the ground?  \nKuta Software-Infinite Calculus Name___________________________________  \nRelated Rates Date________________ Period____  \nSolve each related rate problem.  \n1) Water leaking onto a floor forms a circular pool. The radius of the pool increases at a rate of 4 cm/min. How fast is the area of the pool increasing when the radius is 5 cm?  \nA = area of circle r = radius t = time  \n2  dr   dA Equation: A = πr Given rate: = 4 Find:  \ndt dt  \ndAdt~~ ~~ = 2πr ⋅ ~~ ~~drdt = 40π cm²/min r = 5  \nr = 5  \n2) Oil spilling from a ruptured tanker spreads in a circle on the surface of the ocean. The area of the spill increases at a rate of 9π m²/min. How fast is the radius of the spill increasing when the radius is 10 m?  \nA = area of circle r = radius t = time  \nEquation: A = πr2 Given rate: dAdt = 9π Find: ~~ ~~ddrt  \nr = 10  \ndr  \ndt  \n= 21πr ⋅ dAdt = ~~ ~~920 m/min r = 10  \n3) A conical paper cup is 10 cm tall with a radius of 10 cm. The cup is being filled with water so that the water level rises at a rate of 2 cm/sec. At what rate is water being poured into the cup when the water level is 8 cm?  \nV = volume of material in cone h = height t = time Equation: V = πh33~~ ~~ Given rate: ~~ ~~ddht = 2 Find: ~~ ~~ddVt  \nh = 8  \nddVt~~ ~~ = πh2 ⋅ ddht = 128π cm³/sec  \nh = 8  \n4) A spherical balloon is inflated so that its radius (r) increases at a rate of the volume of the balloon increasing when the radius is 4 cm?  \n2 r  \ncm/sec. How fast is  \nV = volume of sphere r = radius t = time  \n4 3 Equation: V =  3  πr  \nGiven rate: ~~ ~~drdt = ~~ ~~2r  \nFind:  \ndV  \ndt  \nr = 4  \ndV  \ndt  \n2 dr  \n= 4πr ⋅  dt = 32π cm³/sec r = 4  \n5) A 7 ft tall person is walking away from a 20 ft tall lamppost at a rate of 5 ft/sec. Assume the scenario can be modeled with right triangles. At what rate is the length of the person's shadow changing when the person is 16 ft from the lamppost?  \nx = distance from person to lamppost y = length of shadow t = time  \nx + y   y   dx   dy  \nEquation: = Given rate: = 5 Find:  \n20 7 dt dt  \nx = 16  \ndydt~~ ~~ = ~~ ~~713 ⋅ dxdt = ~~ ~~3513 ft/sec x = 16  \n6) An observer stands 700 ft away from a launch pad to observe a rocket launch. The rocket blasts off and maintains a velocity of 900 ft/sec. Assume the scenario can be modeled as a right triangle. How fast is the observer to rocket distance changing when the rocket is 2400 ft from the ground?  \na = altitute of rocket z = distance from observer to rocket t = time  \nEquation: a 2 + 490000 = z 2 Given rate: dd","cbCaicW2jQ20du93","https://ap.wps.com/l/cbCaicW2jQ20du93","pdf",37109,"English","# Related Rates Worksheets\n## Circular Pool Area\n## Oil Spill Radius\n## Conical Cup Filling\n## Spherical Balloon Inflation\n## Person Shadow Length\n## Rocket Distance Change","[{\"question\":\"How do you find the rate of change of area when a circle’s radius increases?\",\"answer\":\"Use A = πr², differentiate with respect to time, and substitute the given dr/dt and the radius value.\"},{\"question\":\"What equation is used for the conical cup volume when the water height rises?\",\"answer\":\"Use V = (1/3)πr²h, relate r to h from similar triangles, differentiate with respect to time, then plug in the given dh/dt and the height.\"},{\"question\":\"How are right-triangle related rates problems set up for the shadow and the rocket?\",\"answer\":\"Define the triangle’s changing legs and hypotenuse, write the Pythagorean relationship (x² + y² = constant forms), differentiate, then solve for the requested rate at the given distance.\"}]","Related Rates - Related Rates | PDF",1791426260]