Problem 460 - Calculus Volume 1

Question

The average cost per gigabyte of RAM can be approximated \(\quad\) by the function \(C(t)=8,500,000(0.65)^{t},\) where \(t\) is measured in years since 1980 , and \(C\) is cost in US\$. Find the average cost per gigabyte of RAM for 1980 to 2010 .

Step-by-Step Solution

Answer

Average cost per gigabyte of RAM from 1980 to 2010 is computed using yearly costs and their average over 31 years.

Step 1: Understand the Function

The function given is \( C(t) = 8,500,000 imes (0.65)^{t} \). This represents the cost \( C \) of RAM per gigabyte as a function of \( t \), where \( t \) is the number of years since 1980. For example, when \( t = 0 \), it represents the year 1980, and when \( t = 30 \), it represents the year 2010.

Step 2: Identify the Range

We need to calculate the average cost for each year from 1980 to 2010. Therefore, \( t \) will range from 0 to 30 inclusive.

Step 3: Set Up the Calculation for Yearly Costs

Calculate the cost for each year from 1980 to 2010, using the function \( C(t) = 8,500,000 imes (0.65)^{t} \). We'll calculate this for each \( t \) from \( 0 \) to \( 30 \).

Step 4: Calculate Cost for Each Year

Calculate \( C(t) \) for \( t = 0 \) to \( t = 30 \). For example, \( C(0) = 8,500,000 \, \text{USD}, \quad C(1) = 8,500,000 imes 0.65 \, \text{USD}, \quad C(2) = 8,500,000 imes (0.65)^2 \, \text{USD} \), and so on.

Step 5: Calculate the Total Cost

Add up all the values of \( C(t) \) for \( t = 0 \) to \( 30 \) to get the total cost over all these years.

Step 6: Compute the Average Cost

Divide the total cost obtained in Step 5 by the number of years (31 years) to calculate the average cost per year for the period from 1980 to 2010.

Step 7: Final Verification and Conclusion

Verify the calculation by double-checking a few values in the provided range for potential errors in computation. Summarize that the average cost per gigabyte from 1980 to 2010 is the desired value.

Key Concepts

Exponential Functions

Exponential functions are powerful tools in mathematics that describe situations where a quantity grows or decays at a consistent relative rate. In our example, the cost of RAM per gigabyte is dictated by the function \( C(t) = 8,500,000 \times (0.65)^{t} \). This specific function shows a decrease in cost over time, as evidenced by the base of the exponent, 0.65, which is less than 1.

• The constant factor 8,500,000 reflects the cost in USD when \( t = 0 \) (or 1980).
• The term \((0.65)^{t}\) describes how cost decreases exponentially as \( t \), the number of years increases.

Understanding exponential decay, like in this example, is crucial because many technological costs exhibit this behavior, depreciating rapidly during initial years.

Average Cost Calculation

Calculating the average cost over a period involves several steps. The goal is to find a balancing point of values over time. First, we calculate the costs for each year using the function \( C(t) = 8,500,000 \times (0.65)^{t} \) from \( t = 0 \) to \( t = 30 \).
Next, we sum these individual costs over the entire time period. This gives us the total cost incurred between 1980 and 2010.
The final step is to divide the total cumulative cost by the number of years, which in this case is 31.
This average gives insight into the general trend of costs over time and how they've adjusted on average per year.

Mathematical Modeling

Mathematical modeling is a key method for representing real-world phenomena via equations. Here, the cost of RAM is modeled with the exponential function \( C(t) = 8,500,000 \times (0.65)^{t} \). Such models help us understand underlying patterns and make informed predictions.

• By modeling the cost, we can see the rate at which technology becomes more accessible over time.
• Mathematical models allow us to experiment with different scenarios without real-world constraints.
• They are essential for planning and forecasting in economics and technology.

In this context, the model provides insights into past trends and the effectiveness of technological progression in reducing costs.

Yearly Cost Analysis

Yearly cost analysis involves the detailed examination of costs over specific periods, here from 1980 to 2010. By investigating year-by-year changes using the equation \( C(t) = 8,500,000 \times (0.65)^{t} \), we can pick up on trends and make data-driven conclusions.

• Each calculated cost for \( t = 0 \) to \( t = 30 \) reflects not just isolated values but part of a broader pattern of exponential decay.
• This analysis can spotlight years with dramatic cost drops, showcasing points where technological advancements were particularly impactful.
• Conducting a comprehensive yearly cost analysis helps in understanding not just averages but also the impact of specific changes over time.

Such analyses are invaluable for businesses and consumers alike, as they provide tangible benchmarks for reviewing past decisions and future planning.