Question

The following problems consider the historic average cost per gigabyte of RAM on a computer. $$\begin{array}{|l|l|} \hline \text { Year } & \text { 5-Year Change (\$) } \\ \hline 1980 & 0 \\ \hline 1985 & -5,468,750 \\ \hline 1990 & -755,495 \\ \hline 1995 & -73,005 \\ \hline 2000 & -29,768 \\ \hline 2005 & -918 \\ \hline 2010 & -177 \\ \hline \end{array}$$ If the average cost per gigabyte of RAM in 2010 is \(\$ 12\), find the average cost per gigabyte of RAM in 1980 .

Short answer

The average cost per gigabyte of RAM in 1980 was $6,328,125.

Step-by-step solution

Identify Given Data

We know from the problem that in 2010, the average cost per gigabyte of RAM is $12. We also know the 5-year change for each time interval starting from 1980 to 2010.

Set Up the Equation

To find the cost per gigabyte of RAM in 1980, we need to work backwards using the 5-year change data. Let's denote the average cost per gigabyte of RAM in 1980 as \( x \).

Calculate Cost for 1985

Starting from 2010 going backwards, add the 5-year change to the known cost in 2010. First, \( 2010 \) back to \( 2005 \): Cost\( = 12 + 177 = 189 \).

Calculate Cost for 2000

Next, work backwards from \( 2005 \) to \( 2000 \): Cost\( = 189 + 918 = 1107 \).

Calculate Cost for 1995

Now, go from \( 2000 \) to \( 1995 \): Cost\( = 1107 + 29,768 = 30,875 \).

Calculate Cost for 1990

Next, from \( 1995 \) to \( 1990 \): Cost\( = 30,875 + 73,005 = 103,880 \).

Calculate Cost for 1985

From \( 1990 \) to \( 1985 \): Cost\( = 103,880 + 755,495 = 859,375 \).

Calculate Cost for 1980

Finally, from \( 1985 \) to \( 1980 \): Cost\( = 859,375 + 5,468,750 = 6,328,125 \). This represents the average cost per gigabyte of RAM in \( 1980 \).

Key concepts

Problem Solving in Calculus

Calculus, often viewed as the mathematical study of change, is a field filled with problem-solving opportunities. Understanding calculus is crucial when dealing with historic cost calculations as seen in this exercise. Here, we have to apply a logical thought process to backwards calculations. This often involves understanding the relationships between data, change over time, and utilizing provided figures effectively.

When dealing with backward calculation, a key calculus concept is comprehension of how functions work over discrete intervals. Each "5-Year Change" provided in the table can be viewed as a piecewise function, standing alone as steps between intervals. By adding these values over different time periods, we retrace our steps.

For effective problem solving in this scenario, visualizing the data and understanding each step's impact is essential. Calculus problems often hinge on recognizing patterns and deducing past values through these established changes.

Vertical Computation

Vertical computation might initially seem daunting, but it is a straightforward approach in many contexts. In this exercise, we use vertical computation to add increments or decrements to known data points. Imagine it like reverse engineering, where each computation builds up to a final figure.

By taking the known cost of RAM in 2010 and backtracking one step at a time, we have effectively used a vertical computation strategy to solve the problem. Each addition gives us the price rollback to a previous year.

The key to solving problems with vertical computation is keeping track of each step clearly. Ensure calculations are carefully recorded, as subsequent computations rely on earlier results being accurate.

Yearly Data Table Analysis

Analyzing data tables effectively is crucial in deriving meaningful information from datasets. Within this exercise, a table has outlined 5-yearly changes in the cost of RAM, providing us with critical steps backwards through time.

Yearly data analysis involves closely examining each piece of data, understanding its significance and how it contributes to the overarching trend. Here, the changes given for each 5-year period show a decreasing trend in pricing, reflecting technological advancements and market changes over time.

Look at each data interval as part of a series that, when unified, tells a historical pricing tale. This helps us understand not only individual data points but also the entire timeframe of analysis. Dissecting each line for its potential adjustments provides the complete historical cost picture, from the year in focus backwards to its origin.

Backward Calculation Method

Backward calculation methods are invaluable in scenarios where the present value is known, and the historical value is needed. This method involves "reverse engineering" known results back to an earlier unknown point, using intermediary change values, like in the exercise provided.

To implement backward calculation, start with the known endpoint (in this example, 2010 at $12) and add each intervening 5-year change sequentially, tracing back to the initial year's value. With clarity and patience, backward calculations reveal earlier values by reversing each prior adjustment, ultimately giving a historic price.

The strength of this method lies in its simplicity and logical consistency. By articulating each step in reverse, you ensure a complete portrayal of progression and gain insights into how values transitioned over time, returning you to the origin of your dataset of interest. Mastering this method provides a reliable tool for uncovering historical data.