Question

counting customers in the grocery store
he food emporium grocery store has two doors for customers to use: an \enter only\
por and an \exit only\ door. sensors in each of the doorways allow the manager of the
ood emporium to keep track of the number of customers inside the store at any given
ne. each day, the store opens at 6:00 a.m. and closes at 11:00 p.m. the manager checks
e customer count several times over the course of one day and makes the following tabl
(a) determine the average rates of change of the number of customers in the sto
from 7:30 a.m. to 9:00 a.m. and from 5:30 p.m. to 9:00 p.m. explain what the
average rates of change over these time intervals mean in the context of the

Explanation:

Step1: Recall the formula for average rate of change

The formula for average rate of change is $\frac{\Delta y}{\Delta x}=\frac{y_2 - y_1}{x_2 - x_1}$, where $y$ is the number of customers and $x$ is the time.

Step2: Calculate the average rate of change from 7:30 a.m. to 9:00 a.m.

First, convert the time to a single - unit format. 7:30 a.m. to 9:00 a.m. is $1.5$ hours. $y_1 = 98$, $y_2=253$.
$\frac{y_2 - y_1}{x_2 - x_1}=\frac{253 - 98}{1.5}=\frac{155}{1.5}=\frac{310}{3}\approx103.33$ customers per hour.
This means that, on average, the number of customers in the store is increasing at a rate of approximately $103.33$ customers per hour from 7:30 a.m. to 9:00 a.m.

Step3: Calculate the average rate of change from 5:30 p.m. to 9:00 p.m.

5:30 p.m. to 9:00 p.m. is $3.5$ hours. $y_1 = 321$, $y_2 = 93$.
$\frac{y_2 - y_1}{x_2 - x_1}=\frac{93 - 321}{3.5}=\frac{- 228}{3.5}=-\frac{456}{7}\approx - 65.14$ customers per hour.
This means that, on average, the number of customers in the store is decreasing at a rate of approximately $65.14$ customers per hour from 5:30 p.m. to 9:00 p.m.

Answer:

The average rate of change from 7:30 a.m. to 9:00 a.m. is approximately $103.33$ customers per hour (customers are increasing). The average rate of change from 5:30 p.m. to 9:00 p.m. is approximately $-65.14$ customers per hour (customers are decreasing).