Problem 25 - Calculus

Question

A photocopy machine can reduce copies to \(80 \%\) of their original size. By copying an already reduced copy, further reductions can be made. (a) If a page is reduced to \(80 \%,\) what percent enlargement is needed to return it to its original size? (b) Estimate the number of times in succession that a page must be copied to make the final copy less than \(15 \%\) of the size of the original.

Step-by-Step Solution

Answer

(a) 25% enlargement is needed. (b) About 14 times reduction is needed.

Step 1: Understanding the initial reduction

Let's assume the original size of the page is 100%. After reducing it to 80%, the new size of the page is 80% of the original size.

Step 2: Calculating the enlargement

To return the page reduced to 80% back to its original size, we need to calculate what percentage increase from the reduced size will bring it back to 100%. If the reduced size is 80%, we set up the equation: \((1 + x) \times 80\% = 100\%\), where \(x\) is the required enlargement percentage.

Step 3: Solving the enlargement equation

Solve the equation \((1 + x) \times 80 = 100\). Divide both sides by 80: \(1 + x = 1.25\). Thus, \(x = 0.25\) or \(25\%\). This means a 25% enlargement is needed.

Step 4: Understanding repeated reductions

Each time the page is copied, it is reduced to 80% of its current size. This is a repeated process of multiplying the current size by 0.8.

Step 5: Setting up the reduction equation

Let's find the number of times (\(n\)) the page needs to be reduced such that its size is less than 15% of the original size. This can be represented by the inequality: \(0.8^n \times 100\% < 15\%\).

Step 6: Solving for \(n\)

Simplify the inequality to \(0.8^n < 0.15\).Take the logarithm of both sides: \(\log(0.8^n) < \log(0.15)\).Using the property of logarithms, this becomes \(n \cdot \log(0.8) < \log(0.15)\).Solve for \(n\) by dividing both sides by \(\log(0.8)\): \(n > \frac{\log(0.15)}{\log(0.8)}\).Calculate this to find \(n \approx 14\).

Step 7: Conclusion

A 25% enlargement is needed to restore the reduced page to its original size, and the page must be reduced about 14 times in succession to get a size that is less than 15% of the original.

Key Concepts

Enlargement Percentage

When you reduce an object, like a photocopy, to a fraction of its original size, it's important to know how to return it to its full size. This involves calculating the enlargement percentage. Let's imagine a page that is reduced to 80% of its original size. To determine how much you need to enlarge this reduced page to bring it back to its original 100% size, you can set up an equation:

• Let the original size be 100%.
• The reduced size is 80% of the original size.
• We need to calculate the percentage increase needed from 80% to 100%.

The formula you'll use is \[(1 + x) \times 80 \%= 100 \%\]where \(x\) is the enlargement percentage. Solving this equation: \[(1 + x) \times 80 = 100\]By dividing both sides by 80, you find \[1 + x = 1.25\]Thus, \[x = 0.25\] or a 25% enlargement. That means to enlarge something reduced by 80% back to its full size, you need a 25% increase.

Repeated Reductions

Understanding repeated reductions is key when dealing with photocopying processes, among other practical applications. Every time a page is copied and reduced to 80% of its initial size, it experiences a repeated reduction. Here's how it works:

• Start with 100% as the original size.
• Each copy results in reduction to 80% of the current size.
• This becomes a multiplicative process, multiplying the current size by 0.8 repeatedly.

To make this concept clearer, consider you're looking to reduce the page until it is less than 15% of its original size. You'd set up this problem using an inequality:\[0.8^n \times 100\% < 15\%\]This equation represents the process of reducing the size to less than 15% through successive use of the photocopy machine.

Logarithmic Inequality

When dealing with repeated reductions, such as making a photocopy successively smaller, logarithmic inequalities help in estimating the number of repetitions needed.To solve the problem of finding the number of times (\(n\)) you need to reduce the page so that it is less than 15% of its original size, you'd use a logarithmic equation:

• Start with: \[0.8^n < 0.15\]
• Apply logarithms to both sides: \[\log(0.8^n) < \log(0.15)\]
• Using the properties of logarithms, rewrite as: \[n \cdot \log(0.8) < \log(0.15)\]
• Then solve for \(n\): \[n > \frac{\log(0.15)}{\log(0.8)}\]

In this conversion, logarithms help estimate the exact number of reductions necessary. Calculating gives \(n \approx 14\), meaning you would need to reduce the size 14 times to get to 15% of the original size.