Problem 52839. Easy Sequences 33: Web Trapped Ant

Created by Ramon Villamangca

Solve Later

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A </w:t></w:r><w:r><w:rPr><w:u/></w:rPr><w:t>spider sitting</w:t></w:r><w:r><w:t> on the </w:t></w:r><w:r><w:rPr><w:u/></w:rPr><w:t>outside</w:t></w:r><w:r><w:t> face of the can, sensed the trapped ant and started crawling towards it. The relative positions of the ant and spider are shown in the figure below.</w:t></w:r></w:p><w:p><w:pPr><w:pStyle w:val=\"text\"/><w:jc w:val=\"left\"/></w:pPr><w:r><w:t> </w:t></w:r><w:customXml w:element=\"image\"><w:customXmlPr><w:attr w:name=\"height\" w:val=\"309\"/><w:attr w:name=\"width\" w:val=\"551\"/><w:attr w:name=\"verticalAlign\" w:val=\"baseline\"/><w:attr w:name=\"altText\" w:val=\"\"/><w:attr w:name=\"relationshipId\" w:val=\"rId1\"/></w:customXmlPr></w:customXml></w:p><w:p><w:pPr><w:pStyle w:val=\"text\"/><w:jc w:val=\"left\"/></w:pPr><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"false\"/></w:customXmlPr><w:r><w:t>hs</w:t></w:r></w:customXml><w:r><w:t> is the initial distance of the spider from the from the bottom of the can, </w:t></w:r><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"false\"/></w:customXmlPr><w:r><w:t>ha</w:t></w:r></w:customXml><w:r><w:t> is the distance of the ant from the top of the can, and </w:t></w:r><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"false\"/></w:customXmlPr><w:r><w:t>xs</w:t></w:r></w:customXml><w:r><w:t> and </w:t></w:r><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"false\"/></w:customXmlPr><w:r><w:t>xa</w:t></w:r></w:customXml><w:r><w:t>, are the y-offsets of the spider and ant, respectively, from the center of the circular top.</w:t></w:r></w:p><w:p><w:pPr><w:pStyle w:val=\"text\"/><w:jc w:val=\"left\"/></w:pPr><w:r><w:rPr><w:b/></w:rPr><w:t>If the spider moves at a constant crawling speed, calculate the total length of the path to reach the ant at the quickest possible time. </w:t></w:r><w:r><w:t>Please round-off your answer to the nearest four decimal places.</w:t></w:r></w:p><w:p><w:pPr><w:pStyle w:val=\"text\"/><w:jc w:val=\"left\"/></w:pPr><w:r><w:rPr><w:b/></w:rPr><w:t>CLARIFICATIONS: </w:t></w:r></w:p><w:p><w:pPr><w:pStyle w:val=\"ListParagraph\"/><w:numPr><w:numId w:val=\"1\"/></w:numPr><w:jc w:val=\"left\"/></w:pPr><w:r><w:t>It is understood there can be 2 possible positions, given a single y-offset. In this exercise please consider only the position as drawn above. That means at the top view, the ant and the spidert shall only be within the first and third quadrant, respectively.</w:t></w:r></w:p><w:p><w:pPr><w:pStyle w:val=\"ListParagraph\"/><w:numPr><w:numId w:val=\"1\"/></w:numPr><w:jc w:val=\"left\"/></w:pPr><w:r><w:t>The dotted lines is not the spider's path. They are used only to project the heights to the 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<div style = "text-align: start; line-height: 20.440000534057617px; min-height: 0px; white-space: normal; color: rgb(0, 0, 0); font-family: Menlo, Monaco, Consolas, monospace; font-style: normal; font-size: 14px; font-weight: normal; text-decoration: none; white-space: normal; "><div style="block-size: 609px; display: block; min-width: 0px; padding-block-start: 0px; padding-top: 0px; vertical-align: baseline; "><div style="block-size: 63px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; text-align: left; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; "><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; ">An <span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; ">ant is<span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; "> trapped on a spider web <span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; ">inside<span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; "> a can with open top. The can has a radius R<span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; "> and height H<span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; ">. A <span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; ">spider sitting<span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; "> on the <span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; ">outside<span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; "> face of the can, sensed the trapped ant and started crawling towards it. The relative positions of the ant and spider are shown in the figure below.</div><div style="block-size: 315px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; text-align: left; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; "><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; "> <img class="imageNode" width="551" height="309" style="vertical-align: baseline;width: 551px;height: 309px" 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src="data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAACEAAAAkCAYAAAAHKVPcAAAAAXNSR0IArs4c6QAAAERlWElmTU0AKgAAAAgAAYdpAAQAAAABAAAAGgAAAAAAA6ABAAMAAAABAAEAAKACAAQAAAABAAAAIaADAAQAAAABAAAAJAAAAADeoA9wAAAClElEQVRYCe2WS4iOYRTHPxNyFzNJFoxEyiW37DQlkwVZUFYuC5E9u1m4RFFKktKwUDZC2VAaUQgbLFyiKMo1lzTNDJOI33+co/M98/a9b1Pft3pP/eY5z7k873me97zPN5VKKeUJlCeQfQLDss0D1hb+zoY5sASWwiG4DA2TizzpPfwJzG/Y08ODRqH3WBEqqC7SlLPqRPzjLOZaTuyQ3XlFrAwr162I8IxM9TRW74mpmRENML62Ih7V81nDayw+C98M83eFuEno62AhXIEbkCVjMW6EueZ8yPgANsER6IVc2UGEv4rVFq0x/WzXZ6yk4t+C8u/Bcfhgc9m82VFry3ncSuiH0aBd/YYLcBjkEycglXMY5HsD3vzT0ftAxRQS3aRfQAtdh3b4ATodyQrwIg4OWKr/vDT/k2pz5STzW4ntf5WpfRGGZjM+ZdTOdkGn2eKne99scfhmk3mM2oDLTZTHPskbdxPgO/2Erk81ym0m8v+CydFh+j5Gz1cPtWTE5JquhkVeoI8JGWqqn+a/E+xRncDks8WomEvRWUQfSZAayHeyKklaE3wdiS9O1ci+hsZt0Zmnt4XkQU2E72jwL85Z7GyI1cnU+tehaqn9IVG7SUWNpZ29SxzTmKv7o6hfPoKfyILorKXrPStJN1rsBeVMAV/wlAzIAdgKbaBe0Z0S5RgTz1HMIGlKLOOZLzdbF+P3xK/7wUX3xxbYDvqFbYURsAGi9IRJod+gtSR41TtDsqt7gv85uk7L74y95nvFqNtR0gz6urRmJxQSHa0XMTMjY3PwqyfaQ8wZ9K9GN+Nd0OvRXaKrPX21mP5J4W71BMZloNeoo+0HFxX0DPTD1Qq6LfUqdOOquFLKEyhPoDyBIZ/AXxJxpWceQHz7AAAAAElFTkSuQmCC" width="16.5" height="18" style="width: 16.5px; height: 18px;"><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; "> is the initial distance of the spider from the from the bottom of the can, <img src="data:image/png;base64,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" width="18" height="18" style="width: 18px; height: 18px;"><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; "> is the distance of the ant from the top of the can, and <img src="data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAACEAAAAkCAYAAAAHKVPcAAAAAXNSR0IArs4c6QAAAERlWElmTU0AKgAAAAgAAYdpAAQAAAABAAAAGgAAAAAAA6ABAAMAAAABAAEAAKACAAQAAAABAAAAIaADAAQAAAABAAAAJAAAAADeoA9wAAACHElEQVRYCe2VzYuOURTAH8Y3JTMhRSgUVkozSbMgjcUMK3Y2kmxYWCnKiiSjSE1W/gExNdkpiqlZTmajsZhJiSQkCsnH71f31O1tZp7nnZ26p36dc8/Hfe577vOct6qKlA6UDpQOlA6UDvyHHVhQc+blxA/BHvgNt+EbhBzGMP4cRsKZdDf6AGyAN/AMumAV3IdGcp2sr+DD/yYG0SE3McLvwVZEAH01xT6j78GjtDb/BjSWHWR66o3gQ9zgJSjnQZ8PmILHELIT4w+Yfzqc6DvJdybztWUOpw3cuA9+gFcxk5zAaZ4cyxK8Fn0HM19bpr88NlbP1dL+LNf3YCGETGBsikW7uoeCOMRr7Pz+W/dajeNLln+pNWG+6yUUfk8bP2ywycWU68F/wd4GNbUp/vIP4KbvarOraik54ynfmkmYq3uE62WIlLgO9bb6kmo3Ob7AUXe2Qc2sKQNEnBX5QU7OkH0B3/4Wf34tD1pijZfryXwPDq41EN+/80FZDHehE57CFcjFCRk1xhvJIrL8xLbDMhiFMdCvvADb63uhzwM8Acf/NHj3HRCyEuMnWHMrnHX6VCqwyJHty7gZQq5hGJO38BGMeyC/Av2XIeQchj6nqxO4kRwnyyL5BPsgl60s4pf5v9CbglvQ1kyB749DaRr0vYK2p6SDydE82ye1jtgRWAsh3v3RtPAK3MP1LoirxCxSOlA6UDowvw78A86/hfND67IrAAAAAElFTkSuQmCC" width="16.5" height="18" style="width: 16.5px; height: 18px;"><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; "> and <img src="data:image/png;base64,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" width="18" height="18" style="width: 18px; height: 18px;"><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; ">, are the y-offsets of the spider and ant, respectively, from the center of the circular top.</div><div style="block-size: 42px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; text-align: left; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; "><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; ">If the spider moves at a constant crawling speed, calculate the total length of the path to reach the ant at the quickest possible time. <span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; ">Please round-off your answer to the nearest four decimal places.</div><div style="block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; text-align: left; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; "><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; ">CLARIFICATIONS: </div><ul style="block-size: 80px; font-family: Helvetica, Arial, sans-serif; list-style-type: square; margin-block-end: 20px; margin-block-start: 10px; margin-bottom: 20px; margin-top: 10px; margin-top: 10px; margin-bottom: 20px; "><li style="block-size: 60px; display: list-item; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-start: 56px; margin-left: 56px; margin-top: 0px; text-align: left; white-space: pre-wrap; margin-left: 56px; ">It is understood there can be 2 possible positions, given a single y-offset. In this exercise please consider only the position as drawn above. That means at the top view, the ant and the spidert shall only be within the first and third quadrant, respectively.</li><li style="block-size: 20px; display: list-item; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-start: 56px; margin-left: 56px; margin-top: 0px; text-align: left; white-space: pre-wrap; margin-left: 56px; ">The dotted lines is not the spider's path. They are used only to project the heights to the side.</li></ul></div></div>

An ant is trapped on a spider web inside a can with open top. The can has a radius  and height . A spider sitting on the outside face of the can, sensed the trapped ant and started crawling towards it. The relative positions of the ant and spider are shown in the figure below.
                         
 is the initial distance of the spider from the from the bottom of the can,  is the distance of the ant from the top of the can, and  and , are the y-offsets of the spider and ant, respectively, from the center of the circular top.
If the spider moves at a constant crawling speed, calculate the total length of the path to reach the ant at the quickest possible time. Please round-off your answer to the nearest four decimal places.
CLARIFICATIONS: 
It is understood there can be 2 possible positions, given a single y-offset. In this exercise please consider only the position as drawn above. That means at the top view, the ant and the spidert shall only be within the first and third quadrant, respectively.
The dotted lines is not the spider's path. They are used only to project the heights to the side.

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Last Solution submitted on Mar 26, 2023

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William on 30 Sep 2021

My results are all different, although the formula gets the right result in all the simple cases like hs=H/2; ha=H/2; xs=0; xa=0; etc...

Ramon Villamangca on 30 Sep 2021

I’m checking my solution…

Ramon Villamangca on 30 Sep 2021

Hi William,

I updated the test suites. I made a mistake in using full diameter instead of the semicircle's diameter. I also added a clarification to ensure only one option is considered.,

With regards to your solution. I think that your algorithm will work only if both the ant and spider are in the same side. Please take note that the ant is inside and the spider is outside. Please try again. Thanks.

Tim on 30 Sep 2021

The dotted line doesn't show it, but presumably the spider has to go up, over the top edge of the can, and down again. Also it can presumably choose whether to go clockwise or counterclockwise.

Ramon Villamangca on 2 Oct 2021

Hi William, I've included one solution into the problem itself. I hope this clarifies everything. Of course the circumference will not change. But if you would not consider inside/outside, the insects will appear to be closer together, and you will be get shorter distance than the correct one.

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Problem 52839. Easy Sequences 33: Web Trapped Ant

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Problem 52839. Easy Sequences 33: Web Trapped Ant

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