Question 1
(a)-Topic-6.1- Exploring Accumulations of Change
(b)-Topic-8.1- Finding the Average Value of a Function on an Interval
(c)-Topic-4.1- Interpreting the Meaning of the Derivative in Context
(d)-Topic-6.5- Interpreting the Behavior of Accumulation Functions Involving Area
1. From 5 A.M. to 10 A.M., the rate at which vehicles arrive at a certain toll plaza is given by \(A(t)=450\sqrt{sin(0.62t)}\), where t is the number of hours after 5 A.M. and A (t) 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. (t = 1) to 10 A.M. (t = 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. (t = 1) to 10 A.M. (t = 5).
(c) Is the rate at which vehicles arrive at the toll plaza at 6 A.M. (t = 1) increasing or decreasing? Give a reason for your answer.
(d) A line forms whenever A ( t) ≥ 400. The number of vehicles in line at time t, for a \(a\leq t\leq 4\), is given by \(N(t)=\int_{a}^{t}(A(x)-400)dx\) , where a is the time when a line first begins to form. To the nearest whole number, find the greatest number of vehicles in line at the toll plaza in the time interval a \(a\leq t\leq 4\). Justify your answer.
▶️Answer/Explanation
1(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. ( t = 1) to 10 A.M. ( t = 5 ).
The total number of vehicles that arrive at the toll plaza from
6 A.M. to 10 A.M. is given by \(\int_{1}^{5}A(t)dt\).
1(b) Find the average value of the rate, in vehicles per hour, at which vehicles arrive at the toll plaza from
6 A.M. ( t = 1) to 10 A.M. ( t = 5 ).
Average =\(\frac{1}{5-1}\int_{1}^{5} A(t) dt\) = 375.536966 5
The average rate at which vehicles arrive at the toll plaza from
6 A.M. to 10 A.M. is 375.537 (or 375.536 ) vehicles per hour.
1(c) Is the rate at which vehicles arrive at the toll plaza at 6 A.M. ( t = 1) increasing or decreasing? Give a reason for your answer.
A′(1) = 148.947272
Because A′(1) > 0, the rate at which the vehicles arrive at the toll plaza is increasing.
1(d) A line forms whenever A(t) ≥ 400. The number of vehicles in line at time t, for a ≤ t≤ 4, is given by \(N(t)=\int_{a}^{t}(A(x)-400)dx\), where a is the time when a line first begins to form. To the nearest whole number, find the greatest number of vehicles in line at the toll plaza in the time interval a ≤ t≤ 4.Justify your answer.
N′(t) = A(t) − 400 = 0
⇒ A(t) = 400 ⇒ = t 1.469372, t = 3.597713
a = 1.469372
b = 3.597713
The greatest number of vehicles in line is 71.