.1. From 5 AM to 10 A.M., the rate at which vehicles arrive at a certain toll plaza is given by A(t) = 450 \u221asin(0.621), where t is the number of hours after 5 A.M. and A() is measured in vehicles per hour. Traffic is flowing smoothly at 5 A.M. with no vehicles waiting in line. (a) Write, but do not evaluate, an integral expression that gives the total number of vehicles that arrive at the toll plaza from 6 A.M. (1 = 1) to 10 A.M. (1 = 5). (b) Find the average value of the rate, in vehicles per hour, at which vehicles arrive at the toll plaza from 6 A.M. (1 = 1) to 10 A.M. (1 = 5). (e) Is the rate at which vehicles arrive at the toll plaza at 6 A.M. (1 = 1) increasing or decreasing?<\/p>\n\n\n\n
The Correct Answer and explanation is :<\/strong><\/mark><\/p>\n\n\n\n To address the problem, we’ll analyze the given rate of vehicle arrival ( A(t) = 450 \\sqrt{\\sin(0.621t)} ) for the hours between 5 A.M. and 10 A.M. Here, ( t ) is measured in hours after 5 A.M., with ( t = 0 ) corresponding to 5 A.M. and ( t = 5 ) corresponding to 10 A.M.<\/p>\n\n\n\n To find the total number of vehicles that arrive at the toll plaza from 6 A.M. (when ( t = 1 )) to 10 A.M. (when ( t = 5 )), we can write the integral expression as follows:<\/p>\n\n\n\n [ This integral represents the total number of vehicles arriving from 6 A.M. to 10 A.M.<\/p>\n\n\n\n The average value of a function ( A(t) ) over the interval ([a, b]) can be calculated using the formula:<\/p>\n\n\n\n [ For our scenario, we will use ( a = 1 ) and ( b = 5 ):<\/p>\n\n\n\n [ To determine whether the rate of vehicle arrival is increasing or decreasing at 6 A.M. (when ( t = 1 )), we can examine the derivative ( A'(t) ).<\/p>\n\n\n\n To find ( A'(t) ), we apply the chain rule:<\/p>\n\n\n\n [ Calculating the derivative:<\/p>\n\n\n\n [ Evaluating this at ( t = 1 ):<\/p>\n\n\n\n Thus, the traffic is becoming denser as the morning progresses, reflecting an increasing rate of vehicle arrivals.<\/p>\n","protected":false},"excerpt":{"rendered":" .1. From 5 AM to 10 A.M., the rate at which vehicles arrive at a certain toll plaza is given by A(t) = 450 \u221asin(0.621), where t is the number of hours after 5 A.M. and A() is measured in vehicles per hour. Traffic is flowing smoothly at 5 A.M. with no vehicles waiting in […]<\/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-162002","post","type-post","status-publish","format-standard","hentry","category-exams-certification"],"_links":{"self":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/162002","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=162002"}],"version-history":[{"count":0,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/162002\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/media?parent=162002"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/categories?post=162002"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/tags?post=162002"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}(a) Integral Expression<\/h3>\n\n\n\n
\\int_{1}^{5} A(t) \\, dt = \\int_{1}^{5} 450 \\sqrt{\\sin(0.621t)} \\, dt
]<\/p>\n\n\n\n(b) Average Value of the Rate<\/h3>\n\n\n\n
\\text{Average value} = \\frac{1}{b – a} \\int_{a}^{b} A(t) \\, dt
]<\/p>\n\n\n\n
\\text{Average value} = \\frac{1}{5 – 1} \\int_{1}^{5} 450 \\sqrt{\\sin(0.621t)} \\, dt = \\frac{1}{4} \\int_{1}^{5} 450 \\sqrt{\\sin(0.621t)} \\, dt
]<\/p>\n\n\n\n(c) Increasing or Decreasing Rate at 6 A.M.<\/h3>\n\n\n\n
A(t) = 450 \\sqrt{\\sin(0.621t)}
]<\/p>\n\n\n\n
A'(t) = 450 \\cdot \\frac{1}{2\\sqrt{\\sin(0.621t)}} \\cdot \\cos(0.621t) \\cdot 0.621
]<\/p>\n\n\n\n\n
Conclusion<\/h3>\n\n\n\n
\n