Problem 358 - Calculus Volume 1

Question

True or False? If true, prove it. If false, find the true answer. Suppose the value of \(\$ 1\) in Japanese yen decreases at 2\(\%\) per year. Starting from \(\$ 1=¥ 250, \quad\) when will \(\$ 1=¥ 1 ?\)

Step-by-Step Solution

Answer

False; it takes approximately 273 years for \$1 to equal ¥1.

Step 1: Understanding the Problem

We need to determine when the value of $1 in Japanese Yen will decrease from ¥250 to ¥1 with a decay rate of 2% per year.

Step 2: Write the Exponential Decay Formula

The value of the Yen after each year can be calculated using the exponential decay formula: \[ V(t) = V_0 \times (1 - r)^t \]where \( V(t) \) is the value after \( t \) years, \( V_0 \) is the initial value (¥250), and \( r \) is the decay rate (0.02 or 2%).

Step 3: Set up the Equation

We need to find \( t \) when \( V(t) = ¥1 \). So, our equation becomes:\[ 1 = 250 \times (1 - 0.02)^t \]

Step 4: Solve for \( t \)

Rearrange the equation to isolate \( t \):\[ (1 - 0.02)^t = \frac{1}{250} \]Take the natural logarithm of both sides:\[ \ln((1 - 0.02)^t) = \ln\left(\frac{1}{250}\right) \]Using the logarithmic identity \( \ln(a^b) = b \cdot \ln(a) \), we get:\[ t \cdot \ln(0.98) = \ln\left(\frac{1}{250}\right) \]\[ t = \frac{\ln\left(\frac{1}{250}\right)}{\ln(0.98)} \]

Step 5: Calculate \( t \) Using a Calculator

Calculate the values using a calculator:\[ \ln\left(\frac{1}{250}\right) \approx -5.521 \\ln(0.98) \approx -0.0202 \ t \approx \frac{-5.521}{-0.0202} \approx 273.2 \]Hence, it will take approximately 273 years for \$1 to equal ¥1.

Step 6: Interpret the Result

Since \( t \approx 273 \), the statement is false. At a 2% annual decrease, it will take about 273 years for \$1 to depreciate to ¥1 under these conditions.

Key Concepts

Exponential Decay Formula

Exponential decay describes the process through which a quantity diminishes at a consistent proportional rate over time. In this context, we're talking about currency depreciation, where the value of a currency decreases consistently year after year. The exponential decay formula can be expressed as:

• \[ V(t) = V_0 \times (1 - r)^t \]

Here, \( V(t) \) is the value after time \( t \) years, \( V_0 \) is the initial value, and \( r \) is the decay rate. This formula helps us compute how much a currency like the Japanese yen will decrease in value over a set period.With an initial value of ¥250 and a decay rate of 2%, you can calculate the value at any future time \( t \) by plugging these into the formula. This predictable pattern of decrease, described by an exponential decay model, provides insights into how values change over time, which is especially useful for financial calculations and forecasts.

Logarithmic Functions

Logarithmic functions are essential when solving equations where the variable is an exponent, such as in the case of exponential decay. To find out how long it takes for ¥250 to decay to ¥1 with a 2% annual decrease, we use logarithms to solve for \( t \), the time variable in the equation:

• \[ 1 = 250 \times (1 - 0.02)^t \]
• To solve for \( t \), rewrite it using logarithms:\[ \ln((1 - 0.02)^t) = \ln\left(\frac{1}{250}\right) \]

Utilizing the logarithmic identity \( \ln(a^b) = b \cdot \ln(a) \), rearrange it to find:

• \[ t \cdot \ln(0.98) = \ln\left(\frac{1}{250}\right) \]

Finally, solve for \( t \) by dividing the logarithms:

• \[ t = \frac{\ln\left(\frac{1}{250}\right)}{\ln(0.98)} \]

Logarithms thus simplify our calculations, allowing us to work backward from the exponential function to find the variable's value.

Currency Depreciation

Currency depreciation refers to the decline in a currency's value in comparison to another. In this exercise, we're tracking how the value of one dollar, initially equivalent to ¥250, decreases over time. The rate of depreciation is crucial because it reveals how quickly a currency reaches a new relative value. Depreciation affects various stakeholders:

• Consumers and businesses dealing in foreign currencies as it impacts the cost of imports and exports.
• Investors who might witness changes in the value of their foreign-denominated assets.

Here, the annual depreciation rate is 2%, meaning the dollar's value drops relative to the yen annually by that percentage. Over long periods, such as 273 years in this scenario, even small annual depreciation rates can significantly alter value, demonstrating the impact of steady, gradual changes over time.

Rate of Change

The rate of change indicates how swiftly a particular quantity is increasing or decreasing. In our example, the quantity in question is the value of currency in yen, and our rate of change is the 2% annual decrease. This rate, while seemingly small, compounds over time, leading to a notable reduction. The importance of understanding the rate of change extends beyond just currency. It applies to:

• Economics, where analysts study changes in market indicators.
• Biology, in populations that grow or shrink at constant relative rates.
• Physics, where decay and growth rates describe many natural processes.

In our mathematical model, the decay rate helps determine how long it will take for the currency value to reach a specific target, ¥1 in this case. Recognizing this rate's compounding nature underscores why long-term changes can have substantial outcomes even if the short-term changes appear minor.