91Ó°ÊÓ

Currency conversion: The number \(P\) of British pounds you can get from a bank is a linear function of the number \(D\) of American dollars you pay. An American tourist arriving at Heathrow airport in England went to a banking window at the airport and gave the teller 70 American dollars. She received 34 British pounds in exchange. In this exercise, assume there is no service charge for exchanging currency. a. What is the rate of change, or slope, of \(P\) with respect to \(D\) ? Explain in practical terms what this number means. (Note: You need two values to calculate a slope, but you were given only one. If you think about it, you know one other value. How many British pounds can you get for zero American dollars?) b. A few days later, the American tourist went to a bank in Plymouth and exchanged 130 American dollars for British pounds. How many pounds did she receive? c. Upon returning to the airport, she found that she still had \(£ 12.32\) in British currency in her purse. In preparation for the trip home, she exchanged that for American dollars. How much money, in American dollars, did she get?

Step by step solution

01

Determine the known point

We know that exchanging 70 American dollars gives 34 British pounds. In point notation, this is (70, 34). We assume that 0 dollars gives 0 pounds, so the other point would be (0, 0). This forms a linear relationship.

02

Calculate the slope (rate of change)

The slope of a line, representing the rate of change, is calculated as \( m = \frac{\Delta P}{\Delta D} = \frac{P_2 - P_1}{D_2 - D_1} \). Using the points (0, 0) and (70, 34), the slope \( m \) is \( \frac{34 - 0}{70 - 0} = \frac{34}{70} = 0.4857 \). This means that for each dollar, you receive approximately 0.4857 British pounds.

03

Interpret the slope

In practical terms, the slope means that for every additional American dollar exchanged, you receive 0.4857 British pounds.

04

Calculate pounds received for 130 dollars

Now that we know the slope, we can use it to find how many pounds can be obtained from 130 dollars. Use the equation of the line \( P = mD \), where \( m \approx 0.4857 \). Substituting \( D = 130 \), we get \( P = 0.4857 \times 130 \approx 63.141 \). So, the tourist received about 63.141 British pounds.

05

Convert pounds to dollars for returning pounds

Using the slope, we need to convert 12.32 British pounds back into American dollars. Rearrange the equation to \( D = \frac{P}{m} \), where \( P = 12.32 \). Substituting, \( D = \frac{12.32}{0.4857} \approx 25.36 \). She received about 25.36 dollars.

Unlock Step-by-Step Solutions & Ace Your Exams!

• Full Textbook Solutions

  Get detailed explanations and key concepts

• Unlimited Al creation

  Al flashcards, explanations, exams and more...

• Ads-free access

  To over 500 millions flashcards

• Money-back guarantee

  We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91Ó°ÊÓ!

Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Understanding Linear Functions

Linear functions represent a relationship between two variables where one variable is directly proportional to the other. In the context of currency conversion, this means that the amount of British pounds you receive (in this case, represented by \( P \)) is a direct function of the American dollars you pay (represented by \( D \)). The graph of this function will always be a straight line, reflecting constant proportionality. This kind of function is useful in expressing consistent relationships like exchange rates, where the rate of conversion stays the same, ensuring predictability in conversions between currencies.

Exploring the Concept of Rate of Change

The rate of change in a linear function is basically how much one variable changes in relation to the change in another variable. In our currency conversion example, it gives us the exchange rate, showing how many pounds you receive for each dollar. Here, the rate of change is the slope, calculated as the change in pounds over the change in dollars.

• In practical terms, this rate tells you how many British pounds you'll get per American dollar.
• It's a key concept in not just currency exchange, but any other setting where you have a constant rate of proportional change between two variables.

Understanding this concept can help you calculate and predict results based on known inputs, making financial planning and spending more reliable.

Calculating the Slope

To find out the rate of conversion from dollars to pounds, the slope of the line formed by this exchange is calculated. The slope formula is \( m = \frac{\Delta P}{\Delta D} \), or more simply, \( m = \frac{P_2 - P_1}{D_2 - D_1} \). For our problem, the points (0,0) and (70,34) are used, where the slope \( m \) equates to \( \frac{34}{70} = 0.4857 \).

• This slope value is crucial as it represents the exchange rate—0.4857 pounds per every dollar exchanged.
• By understanding slope, you can assess how different variables relate, which is essential when dealing with currency conversions or any linear relationship.

The Exchange Rate at Work

Exchange rates reflect the price of trading one currency for another. They can be understood as a math function with a consistent linear relationship. In our problem, the exchange rate is a steady number represented by the slope we calculated earlier, where \( 1 \) US Dollar equals \( 0.4857 \) British Pounds.
Exchange rates affect everything from international travel costs to the study of economies globally, impacting pricing in imports and exports. For travelers, knowing this rate helps effectively manage money and anticipate costs abroad. It signifies consistency, meaning regardless of the amount being exchanged, the relative value remains the same. This predictability is why understanding the calculation of this linear relationship is so important for making informed financial decisions.