Sure, let's solve this step-by-step.
First, we need to determine the formula for the volume of a rectangular box. The volume [tex]\( V \)[/tex] of a rectangular box is given by the product of its length ([tex]\( l \)[/tex]), width ([tex]\( w \)[/tex]), and height ([tex]\( h \)[/tex]). Mathematically, this is:
[tex]\[ V = l \times w \times h \][/tex]
We are given:
- Length ([tex]\( l \)[/tex]) = 20 inches
- Width ([tex]\( w \)[/tex]) = 15 inches
The height ([tex]\( h \)[/tex]) ranges from 4 inches to 6 inches.
Let's calculate the volume for the minimum height ([tex]\( h = 4 \)[/tex] inches):
[tex]\[ V_{\text{min}} = 20 \times 15 \times 4 = 1200 \text{ cubic inches} \][/tex]
Next, let's calculate the volume for the maximum height ([tex]\( h = 6 \)[/tex] inches):
[tex]\[ V_{\text{max}} = 20 \times 15 \times 6 = 1800 \text{ cubic inches} \][/tex]
So, the range of possible volumes for the boxes is from 1200 to 1800 cubic inches.
To visualize this range on a number line:
1. Draw a horizontal line.
2. Mark two points on the line:
- One point at 1200
- Another point at 1800
3. Draw a solid line or a highlighted segment connecting these two points to represent all possible values between 1200 and 1800 cubic inches.
```
|------------------------------------|
1200 1800
```
The highlighted segment between 1200 and 1800 represents the range of possible volumes for the boxes.
