Problem 64 - Algebra 1: Concepts and Skills

Question

A piece of notebook paper is about 0.0032 inch thick. If you begin with a stack consisting of a single sheet and double the stack 25 times, how thick will the stack be? \(H I N T:\) You will need to write and solve an exponential equation.

Step-by-Step Solution

Answer

The thickness of the stack of papers after 25 times of doubling can be found by solving the exponential function \( a = 0.0032 \times 2^{25} \).

Step 1: Understand the Task

There is multiplication by 2, which is a sign of exponential growth. This multiplication situation needs to be turned into an exponential function and solved.

Step 2: Write the Exponential Function

The exponential function will be of the form \( a = p \times b^n \), where:\n- \( a \) is the final amount that results from applying an exponential process to an initial amount \( p \)\n- \( b \) is the base (the factor being applied repeatedly) and \n- \( n \) is the number of times the factor is applied.\nFor this problem:\n- The initial amount of paper (\( p \)) is 0.0032 inch.\n- The factor (\( b \)) is 2 (as the stack is doubled each time).\n- The number of times the stack is doubled (\( n \)) is 25.\nSo, we have: \( a = 0.0032 \times 2^{25} \).

Step 3: Solve the Exponential Function

Plugging in the numbers, we find the thickness of the stack of papers after 25 times of doubling: \( a = 0.0032 \times 2^{25} \).

Key Concepts

Exponential Function

Imagine a process that grows by a constant rate each time it repeats. This process can be described by an exponential function. Such functions are fundamental in mathematics and appear often in real-world scenarios such as population growth, finance, and even folding paper, as in our example.
An exponential function typically takes the form:

• \( a = p \times b^n \)
• where:
  • \( a \) is the final outcome after growth,
  • \( p \) is the initial value you start with,
  • \( b \) is the base or growth factor,
  • \( n \) is the number of times you apply this growth.

In our paper stacking example, each doubling is an application of this function. By understanding how \( b \) and \( n \) interact, you can predict outcomes over time with accuracy.

Exponential Equation

An exponential equation involves expressions where the variable appears in the exponent. This type of equation is essential in solving exponential problems, like in our paper stacking exercise.

The equation reflecting our paper doubling scenario is:

• \( a = 0.0032 \times 2^{25} \)

Here:

• \( 0.0032 \) represents the initial thickness of the paper.
• \( 2 \) is the factor we multiply by each time we double the stack.
• \( 25 \) is the number of doublings.

This equation predicts future outcomes based on a starting point and a consistent growth pattern. Understanding exponential equations helps solve real-world problems where growth or decay occurs exponentially.

Multiplication by 2

Doubling an amount is like repeating a simple process of multiplication by 2. Every time you double something, you multiply the current amount by 2. This process is at the heart of exponential growth.
To visualize this in our stacking paper example, consider how starting with one page changes over time:

• After 1 doubt, you have \( 2^1 = 2 \) pages.
• After 2 doublings, you have \( 2^2 = 4 \) pages.
• After 3 doublings, you have \( 2^3 = 8 \) pages.
• And so on.

Repeating this simple multiplication gives rise to exponential growth. Just like our paper stack becomes incredibly thick after 25 doublings, this principle can scale up small numbers very quickly. Mastering multiplication by 2 helps in understanding the dynamics of exponential changes intuitively.