Problem 34 - Applied Calculus

Question

Soybean production, in millions of tons $$\begin{array}{c|c|c|c|c|c|c} \hline \text { Year } & 2000 & 2001 & 2002 & 2003 & 2004 & 2005 \\ \hline \text { Production } & 161.0 & 170.3 & 180.2 & 190.7 & 201.8 & 213.5 \\ \hline \end{array}$$ 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 needed. (b) 9 successive copies required.

Step 1: Determine Enlargement Factor

A page is reduced to 80% of its original size, so its size becomes 0.8 of the original. To find the enlargement factor needed to return it to original size, solve the equation \( 0.8x = 1 \) where \( x \) represents the enlargement factor.

Step 2: Solve for the Enlargement Factor

Rearrange the equation \( 0.8x = 1 \) to solve for \( x \). Divide both sides by 0.8 to get \( x = \frac{1}{0.8} = 1.25 \). This means an enlargement of 125% is required to return to the original size.

Step 3: Set Up Equation for Successive Reductions

To find out how many times the page needs to be copied to be less than 15% of the original size, use the formula for repeated reductions: \( 0.8^n < 0.15 \), where \( n \) is the number of times the page is copied.

Step 4: Solve for Number of Successive Copies

Take the natural logarithm of both sides: \( \ln(0.8^n) < \ln(0.15) \). Simplify to \( n\ln(0.8) < \ln(0.15) \). Divide both sides by \( \ln(0.8) \), which is negative, so the inequality sign flips: \( n > \frac{\ln(0.15)}{\ln(0.8)} \). Calculate this to get \( n > 8.058 \). Rounding up gives \( n = 9 \).

Key Concepts

Repeated Reductions

When you repeatedly apply the same percentage reduction, it's important to understand how it affects the original size. This process is referred to as repeated reductions. If a photocopy machine reduces a page to 80% of its original size, what actually happens is that each copy produced is 80% the size of the previous one.
This means that, first, the size becomes 0.8 of the initial size. For each successive copy, the new size is again 0.8 of the last copy's size.Think of it like this:

• 1st copy: 0.8 of the original size
• 2nd copy: 0.8 * 0.8 (or 0.8²) of the original size
• 3rd copy: 0.8 * 0.8 * 0.8 (or 0.8³) of the original size

Notice there's a pattern forming. The size of the copy after each reduction is 0.8 raised to the power of the number of copies, denoted as \(0.8^n\).
These reductions continue to decrease the size, and by using this formula, you can estimate how many reductions it takes to reach a desired percentage of the original size.

Enlargement Percentage

An enlargement percentage is needed when you want to return a reduced item to its original size. In this context, if an item shrinks to 80% of its original size, you need to determine the enlargement percentage to bring it back.Here's how to think about it:

• Reduction: Makes the item 80%, or 0.8 times its original size.
• Enlargement: You seek an enlargement factor \(x\) that satisfies \(0.8x = 1\).

To solve this, you rearrange the equation to find \(x\). Divide both sides by 0.8, resulting in \(x = \frac{1}{0.8} = 1.25\). Therefore, the enlargement required is 1.25 times the reduced size, meaning 125% of the reduced size is necessary to reach the original size again.
Understanding this helps you know that a larger percentage, not just 20%, is required to revert back after a reduction to 80%.

Natural Logarithm

The natural logarithm, often denoted as \(\ln\), is a useful tool for solving equations involving exponential relationships, such as those seen in repeated reductions.
In our example, we need to determine how many successive reductions are necessary for the size to be less than 15% of the original. The equation becomes \(0.8^n < 0.15\).To solve this, take the natural logarithm of both sides:

• \(\ln(0.8^n) < \ln(0.15)\)
• This simplifies to \(n\ln(0.8) < \ln(0.15)\)

Since the natural logarithm of a fraction between 0 and 1 is negative, dividing both sides by \(\ln(0.8)\) flips the inequality:

• \(n > \frac{\ln(0.15)}{\ln(0.8)}\)

Calculate this value to determine \(n\). The calculated result suggests that \(n\) must be greater than 8.058 copies, which means you round up to 9. This use of natural logarithms simplifies the process of solving exponential inequalities and understanding growth or decay processes.