What does b, or the vertical intercept, 30 represent in this situation?

Question

x
y
(3, 90)(6, 150)
0
1
2
3
4
5
6
7
8
9
10
0
15
30
45
60
75
90
105
120
135
150
165
180
195
210
Time (minutes)
Amount of Water (liters)
y
=
20x + b
b
=
30

Roger has been filling the tank for 30 minutes.

The amount of water increases 30 liters for every 1 minute.

There are initially 30 liters in the tank when Roger starts filling it.

Answer 1

In the situation you've described, the variable \( b \) represents the y-intercept of the linear equation \( y = 20x + b \). Here, the vertical intercept, or \( b \), is given as 30.

This means that at \( x = 0 \) (which represents the time when Roger starts filling the tank), the amount of water in the tank is 30 liters. In other words, before Roger starts filling the tank (at time 0 minutes), there are already 30 liters of water in the tank.

Thus, to summarize what \( b = 30 \) represents in this context:

• Initial Water in the Tank: There are 30 liters of water in the tank when Roger begins to fill it.
• Starting Point of the Graph: The graph of the equation \( y = 20x + 30 \) intersects the y-axis at the point (0, 30), indicating that the initial amount of water at time zero is 30 liters before any additional water is added.

So, the overall interpretation is that the tank starts with an initial amount of 30 liters of water before Roger starts adding water at the rate of 20 liters per minute.