https://file.aiquickdraw.com/mathbot/file/17273850451450t2vsab3.jpg

The question asks how many pieces of paper towel the company used for a specific number of packages, where each roll contains 64 sheets, and each sheet is made up of 4 layers.

### Key information:
1. Each roll contains 64 sheets.
2. Each sheet is made of 4 layers.
3. The company produces \( 4^{11} \) packages.
4. Each package contains 8 rolls.

### Step-by-step breakdown:
1. **Total number of sheets in one roll**:  
   One roll contains 64 sheets, and each sheet consists of 4 layers, so each roll contains \( 64 \times 4 = 256 \) layers.

2. **Total number of layers in one package**:  
   Since each package contains 8 rolls, the total number of layers in one package is:
   \[
   8 \times 256 = 2048 \text{ layers per package}.
   \]

3. **Total number of packages**:  
   The company produces \( 4^{11} \) packages.

4. **Total number of layers used in all packages**:  
   The total number of layers is:
   \[
   2048 \times 4^{11}.
   \]

5. **Simplify the expression**:
   \[
   2048 = 2^{11}.
   \]
   So, the total number of layers becomes:
   \[
   2^{11} \times 4^{11} = 2^{11} \times (2^2)^{11} = 2^{11} \times 2^{22} = 2^{33}.
   \]

Therefore, the total number of layers used is \( 2^{33} \).

### Final answer:
To match with the answer choices, we express \( 2^{33} \) as:
\[
2^{33} = (2^{11})^3 = (8^{11}) \times 8.
\]
Thus, the total number of layers is **\( 8^{11} \)**, which corresponds to option **C**.

---

Do you want more details or have any questions? Here are 5 related questions:

1. How would the answer change if each roll had 128 sheets instead of 64?
2. What is the impact of doubling the number of layers per sheet?
3. How does the number of layers affect the total weight of paper used?
4. What if the company produced \( 2^{11} \) packages instead of \( 4^{11} \)?
5. How would you calculate the total number of sheets used instead of layers?

**Tip**: When solving exponential problems, try simplifying the base of the powers to make calculations easier.

A paper towel company produces rolls with 64 sheets, each sheet made of 4 layers. The company produces 4^11 packages, with each package containing 8 rolls. How many layers of paper towels are used in total?

Explore how to calculate the total number of layers in paper towel production using exponential rules. This problem involves multiplication, exponentiation, and simplification of expressions, making it suitable for high school students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation