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Answer :
- Substitute $x = 10$ into the equation $y = 20000(0.95)^x$.
- Calculate $y = 20000(0.95)^{10}$, which is approximately $11974.74.
- Compare the result with the given options.
- The closest option is $\boxed{{\$11,975}}$.
### Explanation
1. Understanding the Problem
We are given the equation $y = 20000(0.95)^x$, which models the purchasing power of an initial amount of $20,000 after $x$ years. We want to find the purchasing power 10 years later, so we need to substitute $x = 10$ into the equation.
2. Substituting the Value of x
Substitute $x = 10$ into the equation: $y = 20000(0.95)^{10}$.
3. Calculating the Purchasing Power
Calculate the value of $y$: $y = 20000 \times (0.95)^{10}$. The result of this calculation is approximately $11974.74.
4. Comparing with the Options
Now we compare the calculated value with the given options:
$10,255
$11,975
$12,635
$14,560
The closest value to our calculated result ($11974.74) is $11,975.
5. Final Answer
Therefore, the predicted purchasing power of $20,000 ten years later is approximately $\boxed{{\$11,975}}$.
### Examples
Understanding the depreciation of assets or the erosion of purchasing power due to inflation is crucial in financial planning. For instance, if you have $20,000 today, this equation helps you estimate its value in 10 years, considering a 5% annual decrease in purchasing power. This concept is useful in making informed decisions about investments, savings, and long-term financial goals. It allows you to account for the time value of money and plan accordingly to maintain or grow your wealth effectively.
- Calculate $y = 20000(0.95)^{10}$, which is approximately $11974.74.
- Compare the result with the given options.
- The closest option is $\boxed{{\$11,975}}$.
### Explanation
1. Understanding the Problem
We are given the equation $y = 20000(0.95)^x$, which models the purchasing power of an initial amount of $20,000 after $x$ years. We want to find the purchasing power 10 years later, so we need to substitute $x = 10$ into the equation.
2. Substituting the Value of x
Substitute $x = 10$ into the equation: $y = 20000(0.95)^{10}$.
3. Calculating the Purchasing Power
Calculate the value of $y$: $y = 20000 \times (0.95)^{10}$. The result of this calculation is approximately $11974.74.
4. Comparing with the Options
Now we compare the calculated value with the given options:
$10,255
$11,975
$12,635
$14,560
The closest value to our calculated result ($11974.74) is $11,975.
5. Final Answer
Therefore, the predicted purchasing power of $20,000 ten years later is approximately $\boxed{{\$11,975}}$.
### Examples
Understanding the depreciation of assets or the erosion of purchasing power due to inflation is crucial in financial planning. For instance, if you have $20,000 today, this equation helps you estimate its value in 10 years, considering a 5% annual decrease in purchasing power. This concept is useful in making informed decisions about investments, savings, and long-term financial goals. It allows you to account for the time value of money and plan accordingly to maintain or grow your wealth effectively.
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