Problem 84 - Algebra and Trigonometry

Question

determine whether each statement makes sense or does not make sense, and explain your reasoning. I'm using a photocopier to reduce an image over and over by \(50 \%,\) so the exponential function \(f(x)=\left(\frac{1}{2}\right)^{x}\) models the new image size, where \(x\) is the number of reductions.

Step-by-Step Solution

Answer

The statement makes sense. An image reduced repeatedly by 50% is modeled accurately by the exponential function \(f(x)=\left(\frac{1}{2}\right)^{x}\), where \(x\) is the number of reductions.

Step 1: Understand the scenario

The scenario involves reducing an image repetitively by \(50\%\). This essentially means that each new image id half of the size of the previous one. It is crucial to connect this scenario to exponential decay and determine whether the function \(f(x)=(\frac{1}{2})^{x}\) accurately represents this process.

Step 2: Analyze the function

The function \(f(x)=(\frac{1}{2})^{x}\) is a standard form of an exponential function wherein \( \frac{1}{2}\) (0.5 in decimal form) is the base, and \(x\) represents the exponent which corresponds to the number of times the image has been reduced. For each increment in \(x\), the function value is halved. This aligns with the physical process of the image size reduction.

Step 3: Evaluating the statement

Given the physical process in this scenario matches the mathematical model, we can conclude that the statement 'I'm using a photocopier to reduce an image over and over by \(50 \%,\) so the exponential function \(f(x)=\left(\frac{1}{2}\right)^{x}\) models the new image size, where \(x\) is the number of reductions' makes sense.

Key Concepts

Photocopier Image Reduction

When using a photocopier to reduce an image, you are constantly decreasing the image's size by a certain percentage. In this case, reducing the image by 50% means the new image is half the size of the previous one. This is a repetitive process, so the concept of reducing by half is applied repeatedly with each photocopying pass. Such a reduction is significant because it reflects how the image's dimensions shrink consistently, which can be precisely modeled using mathematical functions.
This shrinking process is consistent and predictable, making it suitable for mathematical modeling. Understanding this basic idea is critical before exploring how exponential decay and mathematical functions come into play.

Exponential Function Modeling

Exponential functions are mathematical expressions that describe processes where a quantity is repeatedly multiplied by a consistent factor. In this scenario, the function \(f(x) = \left(\frac{1}{2}\right)^{x}\) models the image reduction. Here, the base \(\frac{1}{2}\) (or 0.5) represents the factor by which the image size decreases with each photocopying cycle.
With each reduction step \(x\), the image becomes smaller by following the exponential decay's iterative pattern. This is a classic case of exponential function modeling, where the reduction in size operates as a clear and continuous decay modeled by the exponential function.
Understanding the application of exponential models is crucial as they allow predictions about future states based on present conditions. This is seen clearly in repeated processes like photocopier image reduction.

Percent Decrease

Percent decrease is a critical concept when discussing reductions in size over time. In this problem, each photocopied image is 50% smaller than the previous version. It’s a straightforward calculation where each new size is half of what it was, translating to a significant reduction per iteration.
To calculate the size after each reduction, consider it as an ongoing multiplication by 0.5. This approach encapsulates the core principle of a percent decrease - showing how each subsequent state compares to its predecessor.
Understanding percent decrease is essential in exponential decay contexts because it provides an intuitive way to comprehend how quantities diminish continuously and predictably.

Function Analysis

Function analysis involves dissecting a mathematical function to understand its behavior and implications. For \(f(x) = \left(\frac{1}{2}\right)^{x}\), the function expresses the accumulated effect of repetitive reductions. Here, \(x\) signifies the number of reductions applying this inverse growth factor.\Unlike linear functions that change in equal increments, exponential functions like this reflect change by a consistent ratio. This functional behavior allows us to critically analyze the impact of every additional reduction step (each increment in \(x\)).

• Begin by evaluating the starting size, conceptualized as \(f(0) = 1\). This serves as the baseline for reductions.
• Each subsequent \(x\) results in halving the size, offering a clear visualization of exponential decay.

By analyzing this function in detail, you connect the mathematical concepts with real-world processes, deepening your comprehension of exponential decay phenomena in practical applications like photocopying.