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)=0.68d−1
and the function g for exchanging British pounds for Euros is given by the formula
g(p)=1.28p−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 350 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

a) To find the formula for the function that exchanges dollars in return for Euros, we need to combine the two given functions.

First, we need to exchange dollars for British pounds using the function f(d)=0.68d−1. This means we'll receive 0.68d−1 British pounds.

Next, we need to exchange British pounds for Euros using the function g(p)=1.28p−2. Since we have 0.68d−1 British pounds, we can substitute this value into the g(p) function to get the amount of Euros we'll receive:

g(0.68d−1)=1.28(0.68d−1)−2

Simplifying this equation, we get:

g(0.68d−1)=0.872d−2.56−2

Now, we have the formula for the function that exchanges dollars in return for Euros:

Exchange function: h(d) = 0.872d−2.56−2

b) To find how many Euros you would receive for exchanging 350 dollars, substitute d = 350 into the exchange function h(d):

h(350) = 0.872(350)−2.56−2
= 304.4−2.56−2
= 301.84

Therefore, you would receive 301.84 Euros.

c) To find how many dollars you would need to exchange in order to end up with 100 Euros, we need to solve the equation h(d) = 100.

Setting the exchange function h(d) equal to 100, we get:

0.872d−2.56−2 = 100

Simplifying the equation, we have:

0.872d−4.56 = 100

Adding 4.56 to both sides of the equation, we get:

0.872d = 104.56

Dividing both sides by 0.872, we get:

d = 119.93

Therefore, you would need to exchange approximately 119.93 dollars to end up with 100 Euros.