How Tall is an Infinite Stack of Russian Dolls?

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Homework Help Overview

The problem involves a set of Russian Dolls, starting with a first doll that is 1 foot tall, with each subsequent doll being 68.169% the height of its parent. The question posed is about the total height when these dolls are stacked infinitely.

Discussion Character

  • Exploratory, Assumption checking, Mathematical reasoning

Approaches and Questions Raised

  • Participants discuss the nature of the total height of the stacked dolls, with some suggesting it might be infinite while others argue against that notion. There are attempts to formulate expressions for the heights of the dolls and to identify patterns in those expressions.

Discussion Status

The discussion is ongoing, with participants questioning the assumptions about the total height and exploring how to express the heights of the individual dolls. Some guidance has been offered regarding writing expressions for the heights, but no consensus has been reached on the total height.

Contextual Notes

Participants are grappling with the implications of having an infinite number of dolls and how that affects the total height, with some uncertainty about the mathematical expressions needed to describe the situation.

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Homework Statement


Your Russian friend gives you a housewarming gift, and you open it up to find a set of Russian Dolls. The first doll is 1 foot tall. Each time you open one of the wooden dolls you find a smaller doll inside it that is exactly 68.169% the size of its parent. Even when the dolls get very small, there is always another doll inside that is exactly 68.169% the size of the original. You decide to stack the Russian dolls on top of each other and measure how tall the entire set is.

When you add the heights of all the dolls up, you smile and recognize the number instantly!

How tall are the Russian Dolls when they are stacked up on top of each other (you can assume there are an infinite number of nested dolls)?


Homework Equations



Not Sure

The Attempt at a Solution


I assume the number would be infinity
 
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JuliusDarius said:

Homework Statement


Your Russian friend gives you a housewarming gift, and you open it up to find a set of Russian Dolls. The first doll is 1 foot tall. Each time you open one of the wooden dolls you find a smaller doll inside it that is exactly 68.169% the size of its parent. Even when the dolls get very small, there is always another doll inside that is exactly 68.169% the size of the original. You decide to stack the Russian dolls on top of each other and measure how tall the entire set is.

When you add the heights of all the dolls up, you smile and recognize the number instantly!

How tall are the Russian Dolls when they are stacked up on top of each other (you can assume there are an infinite number of nested dolls)?

Homework Equations



Not Sure

The Attempt at a Solution


I assume the number would be infinity
Not true.

The number of dolls is infinite, but the total height is not.

Write an expression for the height of the nth doll.
 
SammyS said:
Not true.

Write an expression for the height of the nth doll.

Would it approach infinity, but never reach it?
 
JuliusDarius said:
Would it approach infinity, but never reach it?

No.

Write an expression for the nth doll
 
micromass said:
No.

Write an expression for the nth doll

Im not sure how to right an expression for the nth doll.
 
Write an expression for the first doll.
Write an expression for the second doll.
Write an expression for the third doll.
Write an expression for the fourth doll.
...

Notice a pattern.
 
micromass said:
write an expression for the first doll.
Write an expression for the second doll.
Write an expression for the third doll.
Write an expression for the fourth doll.
...

Notice a pattern.

1/68.169
2/68.169
3/68.169?
 
I'm not really sure what to write in this situation.
 
JuliusDarius said:
Each time you open one of the wooden dolls you find a smaller doll inside it that is exactly 68.169% the size of its parent.

What's 68.169% of 1 foot (the height of the first doll)? Then what's 68.169% of the height of this doll?
 

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