Question

1. the graph shows the number of visitors at the zoo during different hours of one day. time starts at 9am.

a) find the rate of change in the number of visitors at the zoo between 3pm and 4pm.
describe its meaning in the context of the situation.
b) find the rate of change in the number of visitors
between 5pm and 6pm and describe its meaning in the context of the situation.

Explanation:

Step1: Determine the formula for rate of change

The formula for rate of change (slope) is \(m=\frac{y_2 - y_1}{x_2 - x_1}\), where \((x_1,y_1)\) and \((x_2,y_2)\) are two points on the line.

Step2: Find the points for 3 - 4 pm

Since time starts at 9 am, 3 pm is \(x = 6\) (because \(3+ 3=6\) hours after 9 am) and 4 pm is \(x = 7\). From the graph, at \(x = 6\), \(y_1=150\) and at \(x = 7\), \(y_2 = 200\).

$$m=\frac{200 - 150}{7 - 6}=\frac{50}{1}=50$$

This means that the number of visitors at the zoo is increasing at a rate of 50 visitors per hour between 3 pm and 4 pm.

Step3: Find the points for 5 - 6 pm

5 pm is \(x = 8\) and 6 pm is \(x = 9\). From the graph, at \(x = 8\), \(y_1 = 300\) and at \(x = 9\), \(y_2=0\)

$$m=\frac{0 - 300}{9 - 8}=\frac{- 300}{1}=-300$$

This means that the number of visitors at the zoo is decreasing at a rate of 300 visitors per hour between 5 pm and 6 pm.

Answer:

a) The rate of change is \(50\) visitors per hour. It means the number of visitors is increasing by 50 visitors each hour between 3 pm and 4 pm.
b) The rate of change is \(-300\) visitors per hour. It means the number of visitors is decreasing by 300 visitors each hour between 5 pm and 6 pm.