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 tall will the stack be in inches? How tall will it be in feet? (Hint: Write and solve an exponential equation to find the height of the stack in inches. Then use unit analysis to find the height in feet.)

Short answer

First, to solve for the height in inches, calculate y = 0.0032 * 2^{25}. Then divide the result by 12 to convert the height to feet. The two numerical results will answer the question given.

Step-by-step solution

Calculate the Height in Inches

We start by understanding that the thickness of one sheet of paper is 0.0032 inches and each time we double the stack, the height doubles. Use the formula for exponential growth, \(y = a * b^n\), where a is the initial amount (thickness of the one piece of paper), b is the growth factor (which is 2 since the number of papers is being doubled) and n is the number of times the quantity is being doubled (which is 25 in our case). So, the height of stack in inches will be \(y = 0.0032 * 2^{25}\). Calculate the value of y for the final height in inches.

Convert Inches to Feet

The next step involves converting the height from inches to feet. We know that 1 foot is equal to 12 inches, so to convert inches to feet, the number of inches should be divided by 12. This gives us the final height in feet. So, the final height in feet = \(height \in inches / 12\).

Substitute Values

In the last step, substitute the calculated height of stack in inches in the above formula to find the final height in feet.

Key concepts

Mathematical Modeling

Mathematical modeling forms the basis of solving real-world problems, which can often involve complex phenomena like growth patterns, population dynamics, or, as in our exercise, the stacking of paper sheets. Simplifying these situations into mathematical terms allows us to predict and analyze outcomes using equations. In the case of the notebook paper exercise, the height of a stack doubling 25 times is represented by an exponential equation.

The equation, expressed as \( y = a \times b^n \), captures the essence of mathematical modeling by breaking down the problem into identifiable variables: \( a \) as the initial thickness (0.0032 inches), \( b \) as the growth rate (doubling, thus 2), and \( n \) as the number of iterations (25). By substituting these values and performing calculations, students can predict the final height of the stack—demonstrating how an abstract mathematical formula can provide concrete, real-world solutions.

Unit Conversion

Understanding unit conversion is a fundamental aspect of applied science, mathematics, and daily life activities. The process involves changing the measurement of a quantity without altering its value. When we calculate the stack's height in inches, to bring the number into more comprehensible terms or to fit certain requirements, converting to feet is necessary.

One foot equates to 12 inches, so when we want to convert inches to feet, we simply divide the number of inches by 12. For instance, if we have a stack of paper which measures 3355443.2 inches in height, applying the conversion factor gives us the height in feet: \( 3355443.2 \text{ inches} \/ 12 = 279620.2667 \text{ feet} \). This step is crucial in reporting measurements as it adheres to consistency and standardization in reporting and interpreting results.

Exponential Growth

Exponential growth is an interesting concept that describes a situation where a quantity increases at a rate proportional to its current value. This creates a rapid rise in the quantity over time, which is not just additive—it’s multiplicative. In our exercise, the stack of papers doubles in size each time, growing exponentially.

As you add more layers to the stack, the total size doesn't just increase by the same fixed amount. Instead, each new layer doubles the entire current size, resulting in the height getting very large, very quickly. When this is depicted graphically, the curve starts slowly, then skyrockets upwards—the hallmark of exponential growth. Understanding this concept is key to predicting behaviors in areas such as finance, biology, and, in our example, calculating the seemingly unbelievable height of a doubled stack of paper after 25 iterations.