Suppose you are exchanging currency in the London airport. The currency exchange service there only makes transactions in which one of the two currencies is British pounds, but you want to exchange dollars for Euros. Thus you first need to exchange dollars for British pounds, then exchange British pounds for Euros. At the time you want to make the exchange, the function f for exchanging dollars for British pounds is give by the formula

Question

Suppose you are exchanging currency in the London airport. The currency exchange service there only makes transactions in which one of the two currencies is British pounds, but you want to exchange dollars for Euros. Thus you first need to exchange dollars for British pounds, then exchange British pounds for Euros. At the time you want to make the exchange, the function f for exchanging dollars for British pounds is give by the formula

f(d)=0.7d−1
and the function g for exchanging British pounds for Euros is given by the formula
g(p)=1.27p−2.
(The subtraction of 1 or 2 in the number of British pounds or Euros that you receive is the fee charged by the currency exchange service for each transaction.)
a) Find a formula for the function that exchanges dollars in return for Euros.
Exchange function:

b) How many Euros would you receive for exchanging 150 dollars after going through this two-step exchange process?
Answer: Euros

c) How many dollars would you need to exchange in order to end up with 100 Euros after this two-step exchange process?
Answer: dollars

Answer 1

a) To find a formula for the function that exchanges dollars in return for Euros, we need to think about the process of exchanging dollars for British pounds, and then exchanging British pounds for Euros.

Let's assume the initial amount in dollars is represented by 'd'. The first step is to exchange dollars for British pounds using the function f:
British pounds = f(d) = 0.7d - 1.

Now, we need to exchange the British pounds for Euros using the function g:
Euros = g(British pounds) = g(f(d)) = 1.27(0.7d - 1)^(-2).

Therefore, the formula for the function that exchanges dollars in return for Euros is given by:
Exchange function: h(d) = 1.27(0.7d - 1)^(-2).

b) To find how many Euros you would receive for exchanging 150 dollars, we need to substitute d = 150 into the exchange function h(d):
Euros = h(150) = 1.27(0.7(150) - 1)^(-2).

Calculating this expression will give you the number of Euros you would receive.

c) To find how many dollars you would need to exchange in order to get 100 Euros, we need to set the exchange function h(d) equal to 100 and solve for d:
100 = 1.27(0.7d - 1)^(-2).

This equation can be solved to find the value of d, which represents the number of dollars needed to exchange for 100 Euros.

merge

a) To find a formula for the function that exchanges dollars in return for Euros, we need to think about the process of exchanging dollars for British pounds, and then exchanging British pounds for Euros.