Question

Ravi has a briefcase with dimensions of two by 1 ½ by 1 ½ feet. He wishes to place several notebooks, each eight by nine by one inch, into the briefcase. What is the largest number of notebooks the briefcase will hold?

Hint:

For packing problems, simply dividing the larger volume by the smaller volume can be misleading. Always check the physical fit along each dimension to find the true maximum number.

Answer 1

Step 1: Understanding the Question:
This is a 3D packing problem. We need to find the maximum number of small boxes (notebooks) that can fit inside a larger box (briefcase). We must ensure the units are consistent before calculating.
Step 2: Key Formula or Approach:
1. Convert all dimensions to the same unit (inches).
2. Calculate the volume of the briefcase and the notebook.
3. To find the maximum number, it's best to check how many notebooks fit along each dimension (length, width, height) rather than just dividing volumes, as this accounts for packing orientation.
Step 3: Detailed Explanation:
Part 1: Convert dimensions to inches
Given that 1 foot = 12 inches.
Briefcase dimensions: 2 feet \(\times\) 1.5 feet \(\times\) 1.5 feet.
Length = \(2 \times 12 = 24\) inches.
Width = \(1.5 \times 12 = 18\) inches.
Height = \(1.5 \times 12 = 18\) inches.
So, the briefcase is 24" \(\times\) 18" \(\times\) 18".
Notebook dimensions: 9 inches \(\times\) 8 inches \(\times\) 1 inch.
Part 2: Check packing orientation
We need to find the most efficient way to pack the 9x8x1 notebooks into the 24x18x18 briefcase.
Orientation 1:
- Along the 24-inch dimension, we can fit \(\lfloor \frac{24}{9} \rfloor = 2\) notebooks or \(\lfloor \frac{24}{8} \rfloor = 3\) notebooks. Let's try fitting the 8-inch side. - Number along Length (24"): \(\frac{24}{8} = 3\) notebooks.
- Number along Width (18"): \(\frac{18}{9} = 2\) notebooks.
- Number along Height (18"): \(\frac{18}{1} = 18\) notebooks.
Total notebooks in this orientation = \(3 \times 2 \times 18 = 108\) notebooks.
Orientation 2:
- Let's try fitting the 9-inch side along the 24-inch dimension. - Number along Length (24"): \(\lfloor \frac{24}{9} \rfloor = 2\) notebooks.
- Number along Width (18"): \(\lfloor \frac{18}{8} \rfloor = 2\) notebooks.
- Number along Height (18"): \(\frac{18}{1} = 18\) notebooks.
Total notebooks in this orientation = \(2 \times 2 \times 18 = 72\) notebooks.
Comparing the orientations, the first one yields the largest number.
Step 4: Final Answer:
The largest number of notebooks the briefcase can hold is 108.