a sealed rectnagular box measuring 8*6*18 contains 864 sugar cubes, each measuring 1*1*1. how many sugar cubes are touching the box

Let's break this problem down.

We are given:

- A sealed rectangular box with dimensions \( 8 \times 6 \times 18 \).
- The box contains 864 sugar cubes, each with dimensions \( 1 \times 1 \times 1 \).

We are asked to find how many sugar cubes are touching the box.

### Step 1: Calculate the total number of sugar cubes that touch the box

To solve this, we need to focus on the sugar cubes that are on the surface of the box (i.e., those that are touching one or more faces of the box).

#### Method: Subtract internal sugar cubes from the total.

- The internal sugar cubes are those that are **not** touching any sides of the box. These will be surrounded by other sugar cubes.

#### Step 2: Identify the dimensions of the internal space

The internal cubes will be formed by removing a layer of cubes from each side of the box.

- In the \(8 \times 6 \times 18\) box, if you remove 1 unit from each dimension to account for the outer layer of cubes, the internal dimensions are:
  - Length: \( 8 - 2 = 6 \)
  - Width: \( 6 - 2 = 4 \)
  - Height: \( 18 - 2 = 16 \)

So, the internal box has dimensions \( 6 \times 4 \times 16 \).

#### Step 3: Calculate the number of internal sugar cubes

The number of internal sugar cubes is the volume of this internal space:

\[
6 \times 4 \times 16 = 384 \text{ sugar cubes}.
\]

#### Step 4: Calculate the number of sugar cubes touching the box

The total number of sugar cubes is given as 864. To find the number of sugar cubes touching the box, subtract the internal cubes from the total:

\[
864 - 384 = 480 \text{ sugar cubes}.
\]

### Final Answer:
There are **480 sugar cubes** that are touching the box.

---

Would you like further details on any part of this process? 

Here are some related questions to explore:
1. How would the number of touching cubes change if the box dimensions were different?
2. What is the surface area of the box, and how does that relate to the touching cubes?
3. How many sugar cubes touch exactly one face of the box?
4. How many sugar cubes touch two or more faces of the box?
5. Can we generalize this process for any box dimensions?

**Tip**: For problems involving 3D geometry, visualizing layers and internal structure can often simplify the counting process.

A sealed rectangular box measuring 8*6*18 contains 864 sugar cubes, each measuring 1*1*1. How many sugar cubes are touching the box?

This problem explores how many sugar cubes touch the surface of a sealed rectangular box measuring 8x6x18, containing 864 cubes. Using 3D geometry concepts, the solution covers volume, internal cubes, and surface area. Suitable for middle school students in Grades 6-8.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation