Question
A shipping box is 12 in x 18 in x 16 in. The shipping box will be filled with smaller boxes. Each smaller box is 3 in by 2 in by 4 in. What is the greatest number of smaller boxes that can fit into one shipping box
Answers
GPT-4o mini
To determine the greatest number of smaller boxes that can fit into the shipping box, we first calculate the volume of both the shipping box and the smaller boxes.
1. **Calculate the volume of the shipping box:**
\[
\text{Volume of shipping box} = \text{length} \times \text{width} \times \text{height} = 12 \, \text{in} \times 18 \, \text{in} \times 16 \, \text{in}
\]
\[
= 12 \times 18 = 216 \, \text{in}^2
\]
\[
216 \times 16 = 3456 \, \text{in}^3
\]
Thus, the volume of the shipping box is \( 3456 \, \text{in}^3 \).
2. **Calculate the volume of the smaller box:**
\[
\text{Volume of smaller box} = 3 \, \text{in} \times 2 \, \text{in} \times 4 \, \text{in}
\]
\[
= 3 \times 2 = 6 \, \text{in}^2
\]
\[
6 \times 4 = 24 \, \text{in}^3
\]
Thus, the volume of each smaller box is \( 24 \, \text{in}^3 \).
3. **Calculate the maximum number of smaller boxes that can fit based on volume:**
\[
\text{Maximum number of smaller boxes} = \frac{\text{Volume of shipping box}}{\text{Volume of smaller box}} = \frac{3456 \, \text{in}^3}{24 \, \text{in}^3}
\]
\[
= 144
\]
4. **Confirm fitting based on dimensions:**
Next, we also check how many smaller boxes can fit along each dimension of the shipping box.
- **Along the 12 in dimension:**
\[
\text{Number of smaller boxes} = \frac{12 \, \text{in}}{3 \, \text{in}} = 4
\]
- **Along the 18 in dimension:**
\[
\text{Number of smaller boxes} = \frac{18 \, \text{in}}{2 \, \text{in}} = 9
\]
- **Along the 16 in dimension:**
\[
\text{Number of smaller boxes} = \frac{16 \, \text{in}}{4 \, \text{in}} = 4
\]
5. **Calculate the total number of smaller boxes that can fit:**
\[
\text{Total smaller boxes} = 4 \times 9 \times 4 = 144
\]
So, the greatest number of smaller boxes that can fit into the shipping box is \( \boxed{144} \).
1. **Calculate the volume of the shipping box:**
\[
\text{Volume of shipping box} = \text{length} \times \text{width} \times \text{height} = 12 \, \text{in} \times 18 \, \text{in} \times 16 \, \text{in}
\]
\[
= 12 \times 18 = 216 \, \text{in}^2
\]
\[
216 \times 16 = 3456 \, \text{in}^3
\]
Thus, the volume of the shipping box is \( 3456 \, \text{in}^3 \).
2. **Calculate the volume of the smaller box:**
\[
\text{Volume of smaller box} = 3 \, \text{in} \times 2 \, \text{in} \times 4 \, \text{in}
\]
\[
= 3 \times 2 = 6 \, \text{in}^2
\]
\[
6 \times 4 = 24 \, \text{in}^3
\]
Thus, the volume of each smaller box is \( 24 \, \text{in}^3 \).
3. **Calculate the maximum number of smaller boxes that can fit based on volume:**
\[
\text{Maximum number of smaller boxes} = \frac{\text{Volume of shipping box}}{\text{Volume of smaller box}} = \frac{3456 \, \text{in}^3}{24 \, \text{in}^3}
\]
\[
= 144
\]
4. **Confirm fitting based on dimensions:**
Next, we also check how many smaller boxes can fit along each dimension of the shipping box.
- **Along the 12 in dimension:**
\[
\text{Number of smaller boxes} = \frac{12 \, \text{in}}{3 \, \text{in}} = 4
\]
- **Along the 18 in dimension:**
\[
\text{Number of smaller boxes} = \frac{18 \, \text{in}}{2 \, \text{in}} = 9
\]
- **Along the 16 in dimension:**
\[
\text{Number of smaller boxes} = \frac{16 \, \text{in}}{4 \, \text{in}} = 4
\]
5. **Calculate the total number of smaller boxes that can fit:**
\[
\text{Total smaller boxes} = 4 \times 9 \times 4 = 144
\]
So, the greatest number of smaller boxes that can fit into the shipping box is \( \boxed{144} \).