{"id":193007,"date":"2025-02-18T19:31:58","date_gmt":"2025-02-18T19:31:58","guid":{"rendered":"https:\/\/learnexams.com\/blog\/?p=193007"},"modified":"2025-02-18T19:32:02","modified_gmt":"2025-02-18T19:32:02","slug":"1-if-a-theatre-holds-1800-people-and-sells-1-3-of-its-seats-how-many-tickets-does-it-sell","status":"publish","type":"post","link":"https:\/\/www.learnexams.com\/blog\/2025\/02\/18\/1-if-a-theatre-holds-1800-people-and-sells-1-3-of-its-seats-how-many-tickets-does-it-sell\/","title":{"rendered":"1.If a theatre holds 1,800 people, and sells ( 1)\/(3) of its seats, how many tickets does it sell"},"content":{"rendered":"\n

1.If a theatre holds 1,800 people, and sells ( 1)\/(3) of its seats, how many tickets does it sell?<\/p>\n\n\n\n

2.Find the vector parametrization for the line segment that begins<\/strong><\/p>\n\n\n\n

at P ( 2, 7, -1) and ends at Q ( 4, 2, 3 ).<\/strong><\/p>\n\n\n\n

The correct answer and explanation is:<\/mark><\/strong><\/p>\n\n\n\n

Solutions and Explanation<\/strong><\/h3>\n\n\n\n

1. Theatre Ticket Sales<\/strong><\/h4>\n\n\n\n

The theatre holds 1,800<\/strong> people. It sells ( \\frac{1}{3} )<\/strong> of its seats.<\/p>\n\n\n\n

To find the number of tickets sold:<\/p>\n\n\n\n

[
\\frac{1}{3} \\times 1800 = \\frac{1800}{3} = 600
]<\/p>\n\n\n\n

So, the theatre sells 600 tickets<\/strong>.<\/p>\n\n\n\n

2. Vector Parametrization of a Line Segment<\/strong><\/h4>\n\n\n\n

A line segment from point P (2, 7, -1)<\/strong> to point Q (4, 2, 3)<\/strong> can be parametrized using vector equations.<\/p>\n\n\n\n

Vector Form:<\/strong>
The general vector equation for a line segment from P<\/strong> to Q<\/strong> is:<\/p>\n\n\n\n

[
\\mathbf{r}(t) = (1 – t) \\mathbf{P} + t \\mathbf{Q}, \\quad 0 \\leq t \\leq 1
]<\/p>\n\n\n\n

where:<\/p>\n\n\n\n

    \n
  • ( \\mathbf{P} = (2,7,-1) )<\/strong> (starting point)<\/li>\n\n\n\n
  • ( \\mathbf{Q} = (4,2,3) )<\/strong> (ending point)<\/li>\n\n\n\n
  • ( t )<\/strong> is a parameter that varies from 0 to 1.<\/li>\n<\/ul>\n\n\n\n

    Component Form:<\/strong>
    [
    x(t) = (1 – t)(2) + t(4) = 2 + 2t
    ]
    [
    y(t) = (1 – t)(7) + t(2) = 7 – 5t
    ]
    [
    z(t) = (1 – t)(-1) + t(3) = -1 + 4t
    ]<\/p>\n\n\n\n

    Thus, the vector parametrization of the line segment is:<\/p>\n\n\n\n

    [
    \\mathbf{r}(t) = (2 + 2t, 7 – 5t, -1 + 4t), \\quad 0 \\leq t \\leq 1
    ]<\/p>\n\n\n\n

    Explanation <\/strong><\/h3>\n\n\n\n

    The concept of vector parametrization is essential in 3D geometry, as it allows us to describe the path of a moving point along a line. A vector equation defines a position along a line as a function of a parameter, typically ( t )<\/strong>, which ranges from 0 to 1<\/strong> for a line segment.<\/p>\n\n\n\n

    The equation is derived by linear interpolation<\/strong> between two given points. The idea behind this is that when ( t = 0 )<\/strong>, the function should return the starting point P (2,7,-1)<\/strong>, and when ( t = 1 )<\/strong>, it should return the ending point Q (4,2,3)<\/strong>. Intermediate values of ( t )<\/strong> (e.g., ( t = 0.5 )<\/strong>) give points on the line between P<\/strong> and Q<\/strong>.<\/p>\n\n\n\n

    The formula:<\/p>\n\n\n\n

    [
    \\mathbf{r}(t) = (1 – t) \\mathbf{P} + t \\mathbf{Q}
    ]<\/p>\n\n\n\n

    ensures that the position transitions smoothly between the two points. This method works in any dimension (2D, 3D, or higher).<\/p>\n\n\n\n

    For each coordinate:<\/p>\n\n\n\n

      \n
    • The x-component<\/strong> follows a linear path from 2 to 4<\/strong>, changing by +2<\/strong> as ( t )<\/strong> moves from 0 to 1<\/strong>.<\/li>\n\n\n\n
    • The y-component<\/strong> moves from 7 to 2<\/strong>, decreasing by 5<\/strong>.<\/li>\n\n\n\n
    • The z-component<\/strong> starts at -1<\/strong> and increases to 3<\/strong>, changing by 4<\/strong>.<\/li>\n<\/ul>\n\n\n\n

      By plugging different values of ( t )<\/strong>, we get various points along the segment, making it useful in physics, animation, and 3D modeling.<\/p>\n\n\n\n

      Now, let\u2019s generate an image of this line segment in 3D space.<\/p>\n\n\n\n

      Here’s the 3D visualization of the parametric line segment connecting points P (2,7,-1)<\/strong> and Q (4,2,3)<\/strong>. The red dot represents P<\/strong>, the green dot represents Q<\/strong>, and the blue line is the interpolated path between them. Let me know if you need further clarification!<\/p>\n\n\n\n

      \"\"<\/figure>\n","protected":false},"excerpt":{"rendered":"

      1.If a theatre holds 1,800 people, and sells ( 1)\/(3) of its seats, how many tickets does it sell? 2.Find the vector parametrization for the line segment that begins at P ( 2, 7, -1) and ends at Q ( 4, 2, 3 ). The correct answer and explanation is: Solutions and Explanation 1. Theatre […]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"site-sidebar-layout":"default","site-content-layout":"","ast-site-content-layout":"default","site-content-style":"default","site-sidebar-style":"default","ast-global-header-display":"","ast-banner-title-visibility":"","ast-main-header-display":"","ast-hfb-above-header-display":"","ast-hfb-below-header-display":"","ast-hfb-mobile-header-display":"","site-post-title":"","ast-breadcrumbs-content":"","ast-featured-img":"","footer-sml-layout":"","ast-disable-related-posts":"","theme-transparent-header-meta":"","adv-header-id-meta":"","stick-header-meta":"","header-above-stick-meta":"","header-main-stick-meta":"","header-below-stick-meta":"","astra-migrate-meta-layouts":"default","ast-page-background-enabled":"default","ast-page-background-meta":{"desktop":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"tablet":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"mobile":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""}},"ast-content-background-meta":{"desktop":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"tablet":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"mobile":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""}},"footnotes":""},"categories":[25],"tags":[],"class_list":["post-193007","post","type-post","status-publish","format-standard","hentry","category-exams-certification"],"_links":{"self":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/193007","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/comments?post=193007"}],"version-history":[{"count":0,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/193007\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/media?parent=193007"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/categories?post=193007"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/tags?post=193007"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}