Q. 71 - Calculus

Question

A specialty bookshop is open from 8:00 a.m. to 6:00 p.m. Customers arrive at the shop at a rate r(t) arrivals per hour. A graph of r(t) for the duration of the business day is shown here, where t is the number of hours after the shop opens at 8:00 a.m. As a group, the salespeople in the bookshop can serve customers at a rate of 30 customers per hour. If there are no salespeople available, customers have to wait in line until they can be served, and the customers are willing to wait as long as is necessary in this line. Use the graph ofr(t) to estimate answers to the questions that follow. 

(a) At noon, approximately how many people had entered the bookshop so far? Is there a line yet, and if so, how long is it? 

(b) At what rate is the line at the bookshop growing (or shrinking) at 1:00 p.m.? 

(c) At approximately what time of day is the line the longest, and how many people are in the line at that time?

(d) At 6:00 p.m. when the bookshop closes, has every customer been served or is there still a line at closing time? 

Step-by-Step Solution

Answer

(a) There are 35 customers had served and 5 customers in the shop at noon. 

(b) The rate of arrivals of people at the bookshop is 50 customers per hour. 

(c)  40 people are in the line at 3:00 P.M. 

(d)  There is no customer who is still on the line at the closing time.

Step 1. Given information

A specialty bookshop is open from 8:00 a.m. to 6:00 p.m.  

Customers arrive at the shop at a rate r(t) arrivals per hour.  

A graph of r(t) for the duration of the business day is shown here, where t is the number of hours after the shop opens at 8:00 a.m 

As a group, the salespeople in the bookshop can serve customers at a rate of 30 customers per hour. 

Step 2. (a) At noon, approximately how many people had entered the bookshop so far? Is there a line yet, and if so, how long is it?

The main objective is to calculate the number of people who had entered the bookshop at noon. Find the number of people who are on the line yet. 

Time 8:00 A.M corresponds to the time $t=0$ . Therefore,$t=4$  corresponds to the noon.
At noon, four hours have been spent from opening of the shop. The area under the curve from $t=0$ to $t=4$ indicates number of customers who had entered in the shop.
The area of the triangle with base b and height h is calculated as $A=\frac{1}{2}bh$ .
The base and height of the triangle is 40 and 4 respectively.
Calculate the area of the triangle with base 40 and height 4 .

$$\begin{gathered} A=\frac{1}{2}.40.4 \\ =80 \end{gathered}$$

Therefore, 80 people had entered the bookshop at noon.

It is given that the salesman in the bookshop serves at a rate of 30 customers per hour. Till 11:00 A.M, the rate of arrival is less than 30 . Therefore, there is no line in the bookshop till $t=3$. The rate of arrival of customers exceeded limit with in fourth hour.

The area under the interval $\left[3,4\right]$ gives the number of people who were served by the salesman of bookshop.

The area of trapezium is calculated as $A=\frac{1}{2}(b_1+b_2)h$, where h is the height, b₁ and b₂ are the base of trapezium.
Use the formula of area of trapezium to calculate the area of region on an interval $\left[3,4\right]$.

$$\begin{gathered} A=\frac{1}{2}(40+30).1 \\ =\frac{70}{2} \\ =35 \end{gathered}$$

Therefore, there are 35 customers had served and 5 customers in the shop at noon.

Step 3. (b) At what rate is the line at the bookshop growing (or shrinking) at 1:00 p.m.?

The main objective is to determine the rate of line at the bookshop at 1:00 PM . 

From the given graph, it is observed that the horizontal axis represents the time axis and vertical axis represent the rate $r\left(t\right)$.
In the horizontal axis $t=0$ corresponds to the time 8:00 AM. Therefore, 1:00 PM corresponds to the time $t=5$ .
It is observed form the given graph that the rate of arrivals of people at a bookshop at $t=5$ is 50
Thus, the rate of arrivals of people at the bookshop is 50 customers per hour. 

Step 4. (c) At approximately what time of day is the line the longest, and how many people are in the line at that time?

The main objective is to determine the time in which the line is longest and find the number of people in the line at that time. 

From the given graph, it is observed that the customer arriving rate is increasing till $t=5$. The customers arriving rate decreasing after $t=5$ .
At ,$t=7$ the rate of arrival of customers comes to 30 . Before $t=7$, the rate of arrival of customers is more than 30 and there are more people added to the line.

The area of trapezium is calculated as $A=\frac{1}{2}(b_1+b_2)h$, where h is the height, $b_1$ and $b_2$ are the base of trapezium.
Use the formula of area of trapezium to calculate the area of region on an interval $\left[3,7\right]$ .

$$\begin{gathered} A=\frac{1}{2}(50+30).2+\frac{1}{2}(50+30).2 \\ =\frac{160}{2}+\frac{160}{2} \\ =80+80 \\ =160 \end{gathered}$$

Therefore, there are 160 customers entered the shop in this time.
It is given that the salesman in the bookshop serves at a rate of 30 customers per hour. There are 4 hours on an interval $\left[3,7\right]$.
Multiply 4 by 30 to calculate the number of customers who were served by the salesman. 

$4\times 30=120$

Therefore, 120 customers had served on this time.
Subtract 120 from 160 to calculate the number of customers who are on the line.

$160-120=40$

Thus, 40 people are in the line at 3:00 P.M.

Step 5. (d) At 6:00 p.m. when the bookshop closes, has every customer been served or is there still a line at closing time?

The main objective is to determine the number of customers who are still in the line at 6:00 P.M. 

From part (c), it is clear that 40 people are in the line at 3:00 P.M. After 3:00 P.M, no customer is added in the line because the arrival rate is less than 30 .

The area under the curve from $t=7$ to $t=10$ gives the number of people who arrived in the shop.
Calculate the area under the curve $\left[7,10\right]$.

$$\begin{gathered} A=\frac{1}{2}.30.3 \\ =45 \end{gathered}$$

Therefore, there are $40+45=85$ customers entered in the shop in 3 hours.
It is given that the salesman in the bookshop serves at a rate of 30 customers per hour. There are 3 hours on an interval $\left[7,10\right]$.
Multiply 3 by 30 to calculate the number of customers who were served by the salesman.
$3.30=90$
Thus, there is no customer who is still on the line at the closing time.