The cost of a certain amount of computer storage in 1980 was approximately $250,000. From that time, the cost of the storage has been decreasing at a rate of approximately 33% every 2 years. Find an exponential model for the cost *C* (in dollars) of this amount of storage *t* years after 1980

To create an exponential model for the cost $ C(t) $ of the computer storage $ t $ years after 1980, we start with the initial cost and the rate of decrease.

1. **Initial Cost**: In 1980, the cost $ C(0) $ is $250,000.
2. **Decay Rate**: The cost decreases by approximately 33% every 2 years. This means that after 2 years, the cost will be 67% of its previous value.

### Model Formulation

The exponential decay formula can be expressed as:
\[
C(t) = C_0 \cdot e^{-kt}
\]
However, since we have a discrete decay rate over a specific period (2 years), we can also express it using the form:
\[
C(t) = C_0 \cdot (1 - r)^{t/n}
\]
where:
- $ C_0 = 250,000 $
- $ r = 0.33 $ (the rate of decrease)
- $ n = 2 $ (the period in years over which the rate applies)

### Substituting Values

1. **Finding $ 1 - r $**:
   \[
   1 - r = 1 - 0.33 = 0.67
   \]
   
2. **Final Model**:
   The model becomes:
   \[
   C(t) = 250,000 \cdot (0.67)^{t/2}
   \]

This model indicates that the cost of the computer storage decreases by 33% every 2 years, starting from $250,000 in 1980.

### Summary of the Exponential Model
The exponential model for the cost $ C(t) $ in dollars of the storage $ t $ years after 1980 is:
\[
C(t) = 250,000 \cdot (0.67)^{t/2}
\]

If you have any questions or would like further details, let me know! Here are some related questions you might consider:

1. How would you calculate the cost of storage in 1990 using this model?
2. What would the cost be after 10 years?
3. How many years would it take for the cost to fall below $50,000?
4. How can you graph this model?
5. What implications does this decay have for technology pricing in general?

**Tip**: When modeling decay or growth, always consider the time frame and how frequently the rate applies.

The cost of a certain amount of computer storage in 1980 was approximately $250,000. From that time, the cost of the storage has been decreasing at a rate of approximately 33% every 2 years. Find an exponential model for the cost C (in dollars) of this amount of storage t years after 1980.

Discover the exponential model for the cost of computer storage starting at $250,000 in 1980, decreasing by 33% every 2 years. This page provides a detailed breakdown of the model and related calculations.

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