Problem 461 - Calculus Volume 1

Question

The following problems consider the historic average cost per gigabyte of RAM on a computer. $$\begin{array}{|c|c|c|}\hline \text { Year } & {5 \text { -Year Change (s) }} \\\ \hline 1980 & {0} \\ \hline 1985 & {-5,468,750} \\ \hline 1990 & {-755,495} \\ \hline 1995 & {-73,005} \\ \hline 2000 & {-29,768} \\ \hline 2005 & {-918} \\ \hline 2010 & {-177} \\ \hline\end{array}$$ Find the average cost of 1 \(\mathrm{GB}\) RAM for 2005 to 2010 .

Step-by-Step Solution

Answer

The average cost reduction was $148.2 annually from 2005 to 2010.

Step 1: Understand the Data Table

The table gives the 5-Year Change in the cost of 1 GB of RAM for various years from 1980 to 2010. The value in each row represents how much the cost decreased from the previous year to the given year.

Step 2: Focus on the Relevant Years

The problem is asking for the average cost of 1 GB of RAM specifically from 2005 to 2010. We need to use the change values for 2005 and 2010.

Step 3: Calculate the Total Change from 2005 to 2010

Subtract the 5-year change values for 2010 from that of 2005 to determine the overall change.For 2005: -918, for 2010: -177.The decrease from 2005 to 2010 is calculated by 2005's value minus 2010's value, which is\(-918 - (-177) = -918 + 177 = -741\).

Step 4: Find the Average Cost

Since we're looking for the average over 5 years, divide the total change by 5.\[\text{Average cost} = \frac{-741}{5} = -148.2 \]This suggests an average reduction in cost of $148.2 every year from 2005 to 2010.

Key Concepts

average cost calculation

Average cost calculation involves determining the mean cost of an item or service over a specific period. In this exercise, we're focused on the 5-year average cost change of 1 GB of RAM from 2005 to 2010. Calculating the average cost is a straightforward process:

• First, determine the total change in cost over the period.
• Then, divide this total change by the number of years in the period to find the average yearly change.

For example, if we know the total cost change from 2005 to 2010 is \(-741\), we divide by 5 (number of years) to determine the average annual cost change. Here, it results in the average cost being \(-148.2\) dollars. This average indicates that the cost decreased by approximately $148.2 each year during this time span. Understanding how averages work helps simplify complex data into understandable information, allowing you to gauge general trends over time.

data analysis

Data analysis is about examining, transforming, and arranging data to discover useful information and support decision-making. When looking at changes in RAM prices over time, data analysis involves a few key steps:

• Identify the Data Points: Understand what each data point signifies. For this problem, each year provides a 5-year change in RAM cost.
• Focus on Relevant Data: Concentrate on the years of interest. Here, we are only interested in 2005 and 2010.
• Calculate the Differences: Determine the change in the parameter of interest over the period.

Analyzing the historical RAM price data from 2005 to 2010 shows a decrease by \(-741\). Understanding this data allows us to compute average changes and make sense of price trends. It highlights how past data can reveal patterns and predict future trends.

historical price trends

Historical price trends provide insights into how prices change over time, influenced by technology, supply, demand, and market conditions. In the case of RAM, significant changes have occurred from the 1980s to 2010:

• From 1980 to 2010, there was a continuous decrease in the cost.
• The sharpest decline was observed between the 1980s and 1990s, showing massive advancements in technology and reductions in production costs.
• From 2005 to 2010, although the rate of price decrease slowed, costs continued to decline, reflecting ongoing improvements and efficiencies.

Understanding historical price trends can help anticipate future costs. It is evident that the cost of RAM per GB has been dropping, but each subsequent decade showed varying rates of decrease. Recognizing these trends is crucial for both consumers and businesses, aiding in budget planning and strategic decisions.