{"id":90261,"date":"2025-06-01T10:24:06","date_gmt":"2025-06-01T10:24:06","guid":{"rendered":"https:\/\/exam.pscnotes.com\/mcq\/?p=90261"},"modified":"2025-06-01T10:24:06","modified_gmt":"2025-06-01T10:24:06","slug":"two-pillars-are-placed-vertically-8-feet-apart-the-height-difference","status":"publish","type":"post","link":"https:\/\/exam.pscnotes.com\/mcq\/two-pillars-are-placed-vertically-8-feet-apart-the-height-difference\/","title":{"rendered":"Two pillars are placed vertically 8 feet apart. The height difference"},"content":{"rendered":"

Two pillars are placed vertically 8 feet apart. The height difference of the two pillars is 6 feet. The two ends of a rope of length 15 feet are tied to the tips of the two pillars. The portion of the length of the taller pillar that can be brought in contact with the rope without detaching the rope from the pillars is<\/p>\n

[amp_mcq option1=”less than 6 feet” option2=”more than 6 feet but less than 7 feet” option3=”more than 7 feet but less than 8 feet” option4=”more than 8 feet” correct=”option2″]<\/p>\n

\n
\n
This question was previously asked in<\/div>\n
UPSC CAPF – 2018<\/div>\n<\/div>\n
Download PDF<\/a>Attempt Online<\/a><\/div>\n<\/div>\n
\nThe correct answer is B) more than 6 feet but less than 7 feet.
\n<\/section>\n
\nLet the height of the shorter pillar be $h_1$ and the taller pillar be $h_2$. The distance between the pillars is 8 feet. The height difference is $h_2 – h_1 = 6$ feet. The rope length is 15 feet. The rope is tied to the tips of the pillars.
\nLet the tip of the shorter pillar be A and the tip of the taller pillar be B. Let C be the point on the taller pillar where the rope segment from A first touches the pillar, and the segment from C to B is along the pillar. We are looking for the length of the segment CB, let’s call it $x$.
\nThe coordinates can be set up as A at $(0, h_1)$ and B at $(8, h_2)$. C is on the taller pillar at $(8, y_C)$, where $y_C = h_2 – x$.
\nThe total rope length is the sum of the length of the segment AC and the segment CB.
\nLength of AC = $\\sqrt{(8-0)^2 + (y_C – h_1)^2} = \\sqrt{64 + (h_2 – x – h_1)^2}$.
\nSince $h_2 – h_1 = 6$, this is $\\sqrt{64 + (6 – x)^2}$.
\nLength of CB = $h_2 – y_C = x$.
\nTotal rope length = $\\sqrt{64 + (6 – x)^2} + x = 15$.
\nRearranging the equation: $\\sqrt{64 + (6 – x)^2} = 15 – x$.
\nSquaring both sides: $64 + (6 – x)^2 = (15 – x)^2$
\n$64 + 36 – 12x + x^2 = 225 – 30x + x^2$
\n$100 – 12x = 225 – 30x$
\n$30x – 12x = 225 – 100$
\n$18x = 125$
\n$x = 125 \/ 18$.
\nCalculating the value: $125 \\div 18 \\approx 6.944$ feet.
\nThis value is greater than 6 and less than 7.
\n<\/section>\n
\nThe model assumes the rope is pulled taut along the vertical segment of the taller pillar from the tip downwards. The point C can be at any height on the taller pillar. If $y_C < h_1$, the vertical difference is $h_1 - y_C$. Let $y_C = h_1 - \\Delta y$ where $\\Delta y > 0$. Then $x = h_2 – y_C = h_2 – (h_1 – \\Delta y) = (h_2 – h_1) + \\Delta y = 6 + \\Delta y$. The segment AC length is $\\sqrt{64 + (h_1 – \\Delta y – h_1)^2} = \\sqrt{64 + (-\\Delta y)^2} = \\sqrt{64 + (\\Delta y)^2}$. The total length is $\\sqrt{64 + (\\Delta y)^2} + 6 + \\Delta y = 15$. $\\sqrt{64 + (\\Delta y)^2} = 9 – \\Delta y$. Squaring: $64 + (\\Delta y)^2 = 81 – 18\\Delta y + (\\Delta y)^2$. $64 = 81 – 18\\Delta y$. $18\\Delta y = 17$, $\\Delta y = 17\/18$. Then $x = 6 + \\Delta y = 6 + 17\/18 = (108+17)\/18 = 125\/18$. The result is consistent regardless of whether the point C is above or below the height of the shorter pillar’s tip. The length of contact is approximately 6.944 feet.
\n<\/section>\n","protected":false},"excerpt":{"rendered":"

Two pillars are placed vertically 8 feet apart. The height difference of the two pillars is 6 feet. The two ends of a rope of length 15 feet are tied to the tips of the two pillars. The portion of the length of the taller pillar that can be brought in contact with the rope … <\/p>\n

Detailed SolutionTwo pillars are placed vertically 8 feet apart. The height difference<\/span><\/a><\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"closed","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[1085],"tags":[1114,1102],"class_list":["post-90261","post","type-post","status-publish","format-standard","hentry","category-upsc-capf","tag-1114","tag-quantitative-aptitude-and-reasoning","no-featured-image-padding"],"yoast_head":"\nTwo pillars are placed vertically 8 feet apart. The height difference<\/title>\n<meta name=\"description\" content=\"The correct answer is B) more than 6 feet but less than 7 feet. Let the height of the shorter pillar be $h_1$ and the taller pillar be $h_2$. The distance between the pillars is 8 feet. The height difference is $h_2 - h_1 = 6$ feet. The rope length is 15 feet. The rope is tied to the tips of the pillars. Let the tip of the shorter pillar be A and the tip of the taller pillar be B. Let C be the point on the taller pillar where the rope segment from A first touches the pillar, and the segment from C to B is along the pillar. We are looking for the length of the segment CB, let's call it $x$. The coordinates can be set up as A at $(0, h_1)$ and B at $(8, h_2)$. C is on the taller pillar at $(8, y_C)$, where $y_C = h_2 - x$. The total rope length is the sum of the length of the segment AC and the segment CB. Length of AC = $sqrt{(8-0)^2 + (y_C - h_1)^2} = sqrt{64 + (h_2 - x - h_1)^2}$. Since $h_2 - h_1 = 6$, this is $sqrt{64 + (6 - x)^2}$. Length of CB = $h_2 - y_C = x$. Total rope length = $sqrt{64 + (6 - x)^2} + x = 15$. Rearranging the equation: $sqrt{64 + (6 - x)^2} = 15 - x$. Squaring both sides: $64 + (6 - x)^2 = (15 - x)^2$ $64 + 36 - 12x + x^2 = 225 - 30x + x^2$ $100 - 12x = 225 - 30x$ $30x - 12x = 225 - 100$ $18x = 125$ $x = 125 \/ 18$. Calculating the value: $125 div 18 approx 6.944$ feet. This value is greater than 6 and less than 7.\" \/>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" href=\"https:\/\/exam.pscnotes.com\/mcq\/two-pillars-are-placed-vertically-8-feet-apart-the-height-difference\/\" \/>\n<meta property=\"og:locale\" content=\"en_US\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Two pillars are placed vertically 8 feet apart. The height difference\" \/>\n<meta property=\"og:description\" content=\"The correct answer is B) more than 6 feet but less than 7 feet. Let the height of the shorter pillar be $h_1$ and the taller pillar be $h_2$. The distance between the pillars is 8 feet. The height difference is $h_2 - h_1 = 6$ feet. The rope length is 15 feet. The rope is tied to the tips of the pillars. Let the tip of the shorter pillar be A and the tip of the taller pillar be B. Let C be the point on the taller pillar where the rope segment from A first touches the pillar, and the segment from C to B is along the pillar. We are looking for the length of the segment CB, let's call it $x$. The coordinates can be set up as A at $(0, h_1)$ and B at $(8, h_2)$. C is on the taller pillar at $(8, y_C)$, where $y_C = h_2 - x$. The total rope length is the sum of the length of the segment AC and the segment CB. Length of AC = $sqrt{(8-0)^2 + (y_C - h_1)^2} = sqrt{64 + (h_2 - x - h_1)^2}$. Since $h_2 - h_1 = 6$, this is $sqrt{64 + (6 - x)^2}$. Length of CB = $h_2 - y_C = x$. Total rope length = $sqrt{64 + (6 - x)^2} + x = 15$. Rearranging the equation: $sqrt{64 + (6 - x)^2} = 15 - x$. Squaring both sides: $64 + (6 - x)^2 = (15 - x)^2$ $64 + 36 - 12x + x^2 = 225 - 30x + x^2$ $100 - 12x = 225 - 30x$ $30x - 12x = 225 - 100$ $18x = 125$ $x = 125 \/ 18$. Calculating the value: $125 div 18 approx 6.944$ feet. This value is greater than 6 and less than 7.\" \/>\n<meta property=\"og:url\" content=\"https:\/\/exam.pscnotes.com\/mcq\/two-pillars-are-placed-vertically-8-feet-apart-the-height-difference\/\" \/>\n<meta property=\"og:site_name\" content=\"MCQ and Quiz for Exams\" \/>\n<meta property=\"article:published_time\" content=\"2025-06-01T10:24:06+00:00\" \/>\n<meta name=\"author\" content=\"rawan239\" \/>\n<meta name=\"twitter:card\" content=\"summary_large_image\" \/>\n<meta name=\"twitter:label1\" content=\"Written by\" \/>\n\t<meta name=\"twitter:data1\" content=\"rawan239\" \/>\n\t<meta name=\"twitter:label2\" content=\"Est. reading time\" \/>\n\t<meta name=\"twitter:data2\" content=\"3 minutes\" \/>\n<!-- \/ Yoast SEO Premium plugin. -->","yoast_head_json":{"title":"Two pillars are placed vertically 8 feet apart. The height difference","description":"The correct answer is B) more than 6 feet but less than 7 feet. Let the height of the shorter pillar be $h_1$ and the taller pillar be $h_2$. The distance between the pillars is 8 feet. The height difference is $h_2 - h_1 = 6$ feet. The rope length is 15 feet. The rope is tied to the tips of the pillars. Let the tip of the shorter pillar be A and the tip of the taller pillar be B. Let C be the point on the taller pillar where the rope segment from A first touches the pillar, and the segment from C to B is along the pillar. We are looking for the length of the segment CB, let's call it $x$. The coordinates can be set up as A at $(0, h_1)$ and B at $(8, h_2)$. C is on the taller pillar at $(8, y_C)$, where $y_C = h_2 - x$. The total rope length is the sum of the length of the segment AC and the segment CB. Length of AC = $sqrt{(8-0)^2 + (y_C - h_1)^2} = sqrt{64 + (h_2 - x - h_1)^2}$. Since $h_2 - h_1 = 6$, this is $sqrt{64 + (6 - x)^2}$. Length of CB = $h_2 - y_C = x$. Total rope length = $sqrt{64 + (6 - x)^2} + x = 15$. Rearranging the equation: $sqrt{64 + (6 - x)^2} = 15 - x$. Squaring both sides: $64 + (6 - x)^2 = (15 - x)^2$ $64 + 36 - 12x + x^2 = 225 - 30x + x^2$ $100 - 12x = 225 - 30x$ $30x - 12x = 225 - 100$ $18x = 125$ $x = 125 \/ 18$. Calculating the value: $125 div 18 approx 6.944$ feet. This value is greater than 6 and less than 7.","robots":{"index":"index","follow":"follow","max-snippet":"max-snippet:-1","max-image-preview":"max-image-preview:large","max-video-preview":"max-video-preview:-1"},"canonical":"https:\/\/exam.pscnotes.com\/mcq\/two-pillars-are-placed-vertically-8-feet-apart-the-height-difference\/","og_locale":"en_US","og_type":"article","og_title":"Two pillars are placed vertically 8 feet apart. The height difference","og_description":"The correct answer is B) more than 6 feet but less than 7 feet. Let the height of the shorter pillar be $h_1$ and the taller pillar be $h_2$. The distance between the pillars is 8 feet. The height difference is $h_2 - h_1 = 6$ feet. The rope length is 15 feet. The rope is tied to the tips of the pillars. Let the tip of the shorter pillar be A and the tip of the taller pillar be B. Let C be the point on the taller pillar where the rope segment from A first touches the pillar, and the segment from C to B is along the pillar. We are looking for the length of the segment CB, let's call it $x$. The coordinates can be set up as A at $(0, h_1)$ and B at $(8, h_2)$. C is on the taller pillar at $(8, y_C)$, where $y_C = h_2 - x$. The total rope length is the sum of the length of the segment AC and the segment CB. Length of AC = $sqrt{(8-0)^2 + (y_C - h_1)^2} = sqrt{64 + (h_2 - x - h_1)^2}$. Since $h_2 - h_1 = 6$, this is $sqrt{64 + (6 - x)^2}$. Length of CB = $h_2 - y_C = x$. Total rope length = $sqrt{64 + (6 - x)^2} + x = 15$. Rearranging the equation: $sqrt{64 + (6 - x)^2} = 15 - x$. Squaring both sides: $64 + (6 - x)^2 = (15 - x)^2$ $64 + 36 - 12x + x^2 = 225 - 30x + x^2$ $100 - 12x = 225 - 30x$ $30x - 12x = 225 - 100$ $18x = 125$ $x = 125 \/ 18$. Calculating the value: $125 div 18 approx 6.944$ feet. This value is greater than 6 and less than 7.","og_url":"https:\/\/exam.pscnotes.com\/mcq\/two-pillars-are-placed-vertically-8-feet-apart-the-height-difference\/","og_site_name":"MCQ and Quiz for Exams","article_published_time":"2025-06-01T10:24:06+00:00","author":"rawan239","twitter_card":"summary_large_image","twitter_misc":{"Written by":"rawan239","Est. reading time":"3 minutes"},"schema":{"@context":"https:\/\/schema.org","@graph":[{"@type":"WebPage","@id":"https:\/\/exam.pscnotes.com\/mcq\/two-pillars-are-placed-vertically-8-feet-apart-the-height-difference\/","url":"https:\/\/exam.pscnotes.com\/mcq\/two-pillars-are-placed-vertically-8-feet-apart-the-height-difference\/","name":"Two pillars are placed vertically 8 feet apart. The height difference","isPartOf":{"@id":"https:\/\/exam.pscnotes.com\/mcq\/#website"},"datePublished":"2025-06-01T10:24:06+00:00","dateModified":"2025-06-01T10:24:06+00:00","author":{"@id":"https:\/\/exam.pscnotes.com\/mcq\/#\/schema\/person\/5807dafeb27d2ec82344d6cbd6c3d209"},"description":"The correct answer is B) more than 6 feet but less than 7 feet. Let the height of the shorter pillar be $h_1$ and the taller pillar be $h_2$. The distance between the pillars is 8 feet. The height difference is $h_2 - h_1 = 6$ feet. The rope length is 15 feet. The rope is tied to the tips of the pillars. Let the tip of the shorter pillar be A and the tip of the taller pillar be B. Let C be the point on the taller pillar where the rope segment from A first touches the pillar, and the segment from C to B is along the pillar. We are looking for the length of the segment CB, let's call it $x$. The coordinates can be set up as A at $(0, h_1)$ and B at $(8, h_2)$. C is on the taller pillar at $(8, y_C)$, where $y_C = h_2 - x$. The total rope length is the sum of the length of the segment AC and the segment CB. Length of AC = $\\sqrt{(8-0)^2 + (y_C - h_1)^2} = \\sqrt{64 + (h_2 - x - h_1)^2}$. Since $h_2 - h_1 = 6$, this is $\\sqrt{64 + (6 - x)^2}$. Length of CB = $h_2 - y_C = x$. Total rope length = $\\sqrt{64 + (6 - x)^2} + x = 15$. Rearranging the equation: $\\sqrt{64 + (6 - x)^2} = 15 - x$. Squaring both sides: $64 + (6 - x)^2 = (15 - x)^2$ $64 + 36 - 12x + x^2 = 225 - 30x + x^2$ $100 - 12x = 225 - 30x$ $30x - 12x = 225 - 100$ $18x = 125$ $x = 125 \/ 18$. Calculating the value: $125 \\div 18 \\approx 6.944$ feet. This value is greater than 6 and less than 7.","breadcrumb":{"@id":"https:\/\/exam.pscnotes.com\/mcq\/two-pillars-are-placed-vertically-8-feet-apart-the-height-difference\/#breadcrumb"},"inLanguage":"en-US","potentialAction":[{"@type":"ReadAction","target":["https:\/\/exam.pscnotes.com\/mcq\/two-pillars-are-placed-vertically-8-feet-apart-the-height-difference\/"]}]},{"@type":"BreadcrumbList","@id":"https:\/\/exam.pscnotes.com\/mcq\/two-pillars-are-placed-vertically-8-feet-apart-the-height-difference\/#breadcrumb","itemListElement":[{"@type":"ListItem","position":1,"name":"Home","item":"https:\/\/exam.pscnotes.com\/mcq\/"},{"@type":"ListItem","position":2,"name":"UPSC CAPF","item":"https:\/\/exam.pscnotes.com\/mcq\/category\/upsc-capf\/"},{"@type":"ListItem","position":3,"name":"Two pillars are placed vertically 8 feet apart. The height difference"}]},{"@type":"WebSite","@id":"https:\/\/exam.pscnotes.com\/mcq\/#website","url":"https:\/\/exam.pscnotes.com\/mcq\/","name":"MCQ and Quiz for Exams","description":"","potentialAction":[{"@type":"SearchAction","target":{"@type":"EntryPoint","urlTemplate":"https:\/\/exam.pscnotes.com\/mcq\/?s={search_term_string}"},"query-input":"required name=search_term_string"}],"inLanguage":"en-US"},{"@type":"Person","@id":"https:\/\/exam.pscnotes.com\/mcq\/#\/schema\/person\/5807dafeb27d2ec82344d6cbd6c3d209","name":"rawan239","image":{"@type":"ImageObject","inLanguage":"en-US","@id":"https:\/\/exam.pscnotes.com\/mcq\/#\/schema\/person\/image\/","url":"https:\/\/secure.gravatar.com\/avatar\/761a7274f9cce048fa5b921221e7934820d74514df93ef195a9d22af0c1c9001?s=96&d=mm&r=g","contentUrl":"https:\/\/secure.gravatar.com\/avatar\/761a7274f9cce048fa5b921221e7934820d74514df93ef195a9d22af0c1c9001?s=96&d=mm&r=g","caption":"rawan239"},"sameAs":["https:\/\/exam.pscnotes.com"],"url":"https:\/\/exam.pscnotes.com\/mcq\/author\/rawan239\/"}]}},"amp_enabled":true,"_links":{"self":[{"href":"https:\/\/exam.pscnotes.com\/mcq\/wp-json\/wp\/v2\/posts\/90261","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/exam.pscnotes.com\/mcq\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/exam.pscnotes.com\/mcq\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/exam.pscnotes.com\/mcq\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/exam.pscnotes.com\/mcq\/wp-json\/wp\/v2\/comments?post=90261"}],"version-history":[{"count":0,"href":"https:\/\/exam.pscnotes.com\/mcq\/wp-json\/wp\/v2\/posts\/90261\/revisions"}],"wp:attachment":[{"href":"https:\/\/exam.pscnotes.com\/mcq\/wp-json\/wp\/v2\/media?parent=90261"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/exam.pscnotes.com\/mcq\/wp-json\/wp\/v2\/categories?post=90261"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/exam.pscnotes.com\/mcq\/wp-json\/wp\/v2\/tags?post=90261"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}