Problem 56 - Calculus Volume 1

Question

[T] An American tourist visits Paris and must convert U.S. dollars to Euros, which can be done using the function \(E(x)=0.79 x,\) where \(x\) is the number of U.S. dollars and \(E(x)\) is the equivalent number of Euros. Since conversion rates fluctuate, when the tourist returns to the United States 2 weeks later, the conversion from Euros to U.S. dollars is \(D(x)=1.245 x,\) where \(x\) is the number of Euros and \(D(x)\) is the equivalent number of U.S. dollars. a. Find the composite function that converts directly from U.S. dollars to U.S. dollars via Euros. Did this tourist lose value in the conversion process? b. Use (a) to determine how many U.S. dollars the tourist would get back at the end of her trip if she converted an extra \(\$ 200\) when she arrived in Paris.

Step-by-Step Solution

Answer

The tourist would get back approximately $196.71, losing about 1.645% in the process.

Step 1: Understanding Composite Functions

A composite function applies one function to the results of another. Here, we compose the functions representing currency conversion to go from U.S. dollars to Euros and then back to U.S. dollars.

Step 2: Define the Functions

We are given two functions: the conversion from US dollars to Euros as \(E(x) = 0.79x\) and the conversion from Euros back to US dollars as \(D(x) = 1.245x\).

Step 3: Formulate the Composite Function for Conversion

The composite function is \(D(E(x))\), which combines \(E(x)\) and \(D(x)\). We apply \(E(x)\) first and then \(D(x)\) to that result.\[D(E(x)) = D(0.79x) = 1.245(0.79x)\]

Step 4: Calculate the Composite Function

Simplify the expression for the composite function:\[D(E(x)) = 1.245 \times 0.79x = 0.98355x\]

Step 5: Analyze the Tourist's Loss

The composite function \(D(E(x)) = 0.98355x\) indicates that for each US dollar spent, the tourist receives back 0.98355 dollars. Thus, the tourist loses approximately 1.645% of the value due to conversion.

Step 6: Determine Amount Received for an Extra $200

To find out how many dollars the tourist gets back from their extra \(\$200\), we calculate:\[D(E(200)) = 0.98355 \times 200 = 196.71\]

Key Concepts

Currency Conversion

Currency conversion is the process of exchanging one currency for another. This important financial operation is common for tourists, businesses, and international transactions.
In our exercise, the conversion process involves two steps with two distinct functions:

• First, converting U.S. dollars to Euros using the function \(E(x) = 0.79x\).
• Second, converting back from Euros to U.S. dollars with the function \(D(x) = 1.245x\).

Each function represents a specific rate at the time of the transaction. Understanding these rates is crucial for calculating exactly how much you will have post-conversion.
By knowing these conversions, people can make informed decisions and plan better for travel or business internationally.

Loss in Currency Exchange

When dealing with currency exchange, there is often a loss in value. This is due to changes in exchange rates or service fees applied during conversion.
In the given scenario, the composite function \(D(E(x)) = 0.98355x\) demonstrates that the tourist ends up with less than they initially had after converting their money back to U.S. dollars.

• The calculated equation reveals a loss of approximately \( 1.645\% \) for every dollar converted through the cycle of U.S. dollars to Euros and back.

This situation highlights the importance of being aware of potential losses when exchanging currencies.
Even small percentage changes can add up, leading to significant differences over large transactions. It is crucial to monitor and consider these potential losses whenever dealing with multiple currencies.

Function Composition

Function composition in mathematics involves combining two functions to produce a single function. Here, this concept is applied to model currency conversions.
To find the composite function, we used the functions \(E(x) = 0.79x\) and \(D(x) = 1.245x\).

• The composition is expressed as \(D(E(x))\), which means we first apply \(E(x)\) and then \(D(x)\) to the result.

This yields \[D(E(x)) = 1.245(0.79x) = 0.98355x\].
This process creates a succinct formula that tells us the result of two conversions in one step.
Function composition not only simplifies calculations but also provides a deeper understanding of how inputs and outputs are transformed through linked operations. By mastering this concept, you gain a robust tool for solving complex, multi-step problems in various fields beyond currency conversion.