A hungry bookworm bores through a complete set of encyclopedias consisting of n volumes stacked in numerical order on a library shelf. The bookworm starts inside the front cover of volume​ 1, bores from page 1 of volume 1 to the last page of the last​ volume, and stops inside the back cover of the last volume. Note that the book worm starts inside the front cover of volume 1 and ends inside the back cover of volume n.
Assume that each volume has the same number of pages. For each​ book, assume that you know how thick the cover​ is, and that the thickness of a front cover is equal to the thickness of a back​ cover; assume also that you know the total thickness of all the pages in the book. How far does the bookworm​ travel? How far will it travel if there are 1414 volumes in the set and each book has 33 inches of pages and a  33​/8-inch thick​ cover?
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Part 1
The bookworm will travel enter your response here inches. ​(Type an integer or decimal rounded to two decimal places as​ needed.)

To determine how far the bookworm travels, let's break down the situation step-by-step:

### Key Assumptions:
- The bookworm starts inside the front cover of Volume 1 and stops inside the back cover of Volume \(n\).
- Each volume has a front cover, a back cover, and a certain number of pages.
- The front cover's thickness is equal to the back cover's thickness.
- We are given the following specific values:
  - There are \( n = 1414 \) volumes.
  - Each book has 33 inches of pages.
  - The thickness of each cover is \( \frac{3}{8} \) inches (for both front and back).

### Step-by-Step Breakdown:

1. **Understanding the bookworm's path**:
   - The bookworm starts just inside the front cover of the first volume, bores through all the pages, and finishes just inside the back cover of the last volume. 
   - Since the bookworm does not travel through the outside of the first and last cover, we do not need to count the thickness of the front cover of the first book and the back cover of the last book.

2. **Thickness of one book**:
   - Each book consists of the thickness of its pages plus the thickness of both its front and back covers.
   - Total thickness of one book \(T\) is:
     \[
     T = \text{Pages thickness} + 2 \times \text{Cover thickness} = 33 + 2 \times \frac{3}{8} = 33 + 0.75 = 33.75 \text{ inches}.
     \]

3. **Distance the bookworm travels**:
   - The bookworm skips the front cover of the first volume and the back cover of the last volume. Hence, the distance traveled is:
     \[
     \text{Distance traveled} = (\text{Total thickness of all volumes}) - (\text{Front cover of first volume}) - (\text{Back cover of last volume}).
     \]
   - Since the thickness of each cover is \( \frac{3}{8} \), the total distance traveled becomes:
     \[
     \text{Distance traveled} = 1414 \times 33.75 - \frac{3}{8} - \frac{3}{8}.
     \]
     Simplifying the above:
     \[
     \text{Distance traveled} = 1414 \times 33.75 - 0.75 = 47797.5 - 0.75 = 47796.75 \text{ inches}.
     \]

Thus, the bookworm will travel approximately **47796.75 inches**.

Would you like more details or further explanations?

Here are 5 related questions:
1. How would the result change if each book had a different number of pages?
2. What if the covers were of unequal thickness?
3. How does the total thickness change if we increase the number of volumes to 2000?
4. How does removing the first and last cover from the calculation affect the final result?
5. How would the problem differ if the bookworm only bored through the pages and not the covers?

**Tip**: Breaking a complex problem into smaller parts often simplifies the calculation!

A hungry bookworm bores through a complete set of encyclopedias consisting of n volumes stacked in numerical order on a library shelf. The bookworm starts inside the front cover of volume​ 1, bores from page 1 of volume 1 to the last page of the last​ volume, and stops inside the back cover of the last volume. Note that the book worm starts inside the front cover of volume 1 and ends inside the back cover of volume n. Assume that each volume has the same number of pages. For each​ book, assume that you know how thick the cover​ is, and that the thickness of a front cover is equal to the thickness of a back​ cover; assume also that you know the total thickness of all the pages in the book. How far does the bookworm​ travel? How far will it travel if there are 1414 volumes in the set and each book has 33 inches of pages and a 33​/8-inch thick​ cover?

Learn how to calculate the distance a bookworm travels through 1414 encyclopedia volumes. This problem involves arithmetic and geometric reasoning, using the total thickness of books and cover thicknesses. The solution explains the step-by-step process and provides a formula-based approach. Ideal for students in Grades 10-12.

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