.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 √sin(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?
The Correct Answer and explanation is :
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.
(a) Integral Expression
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:
[
\int_{1}^{5} A(t) \, dt = \int_{1}^{5} 450 \sqrt{\sin(0.621t)} \, dt
]
This integral represents the total number of vehicles arriving from 6 A.M. to 10 A.M.
(b) Average Value of the Rate
The average value of a function ( A(t) ) over the interval ([a, b]) can be calculated using the formula:
[
\text{Average value} = \frac{1}{b – a} \int_{a}^{b} A(t) \, dt
]
For our scenario, we will use ( a = 1 ) and ( b = 5 ):
[
\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
]
(c) Increasing or Decreasing Rate at 6 A.M.
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) ).
To find ( A'(t) ), we apply the chain rule:
[
A(t) = 450 \sqrt{\sin(0.621t)}
]
Calculating the derivative:
[
A'(t) = 450 \cdot \frac{1}{2\sqrt{\sin(0.621t)}} \cdot \cos(0.621t) \cdot 0.621
]
Evaluating this at ( t = 1 ):
- Calculate ( \sin(0.621) ) and ( \cos(0.621) ).
- Since ( A'(t) ) depends on ( \cos(0.621) ), and knowing that the cosine function is positive in the first quadrant (and ( 0.621 ) radians is in this range), we can conclude that ( A'(1) > 0 ).
Conclusion
- The integral expression for the total number of vehicles is given by ( \int_{1}^{5} 450 \sqrt{\sin(0.621t)} \, dt ).
- The average arrival rate can be calculated using the average value formula.
- Since ( A'(1) > 0 ), the rate at which vehicles arrive at the toll plaza is increasing at 6 A.M., indicating that more vehicles are arriving as time progresses from that point.
Thus, the traffic is becoming denser as the morning progresses, reflecting an increasing rate of vehicle arrivals.