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)=0.69d−1

and the function gg for exchanging British pounds for Euros is given by the formula

g(p)=1.3p−2

Required:
a. Find a formula for the function that exchanges dollars in return for Euros.
b. How many Euros would you receive for exchanging 450 dollars after going through this two-step exchange process?
c. How many dollars would you need to exchange in order to end up with 60 Euros after this two-step exchange process?

The function that converts dollars to Euro is [tex]\mathbf{g(d) = 0.897d - 3.3 }[/tex]
You would receive 400.35 Euros, when you exchange 450 dollars
You would receive 70.6 dollars, when you exchange 60 Euros

Converting from dollars to British pounds, we have:

[tex]\mathbf{f(d) = 0.69d - 1}[/tex]

Converting from British pounds to Euro, we have:

[tex]\mathbf{g(p) = 1.3p - 2}[/tex]

(a) Function that converts dollars to Euro

To do this, we simply calculate gf(d)

So, we have:

[tex]\mathbf{g(f(d)) = 1.3f(d) - 2}[/tex]

Substitute [tex]\mathbf{f(d) = 0.69d - 1}[/tex]

[tex]\mathbf{g(f(d)) = 1.3 \times (0.69d - 1) - 2}[/tex]

Expand

[tex]\mathbf{g(f(d)) = 0.897d - 1.3 - 2}[/tex]

[tex]\mathbf{g(f(d)) = 0.897d - 3.3 }[/tex]

Rewrite as:

[tex]\mathbf{g(d) = 0.897d - 3.3 }[/tex]

Hence, the function that converts dollars to Euro is [tex]\mathbf{g(d) = 0.897d - 3.3 }[/tex]

(b) Euro equivalent of $450

In (a), we have [tex]\mathbf{g(d) = 0.897d - 3.3 }[/tex]

Substitute 450 for d

[tex]\mathbf{g(450) = 0.897 \times 450 - 3.3 }[/tex]

[tex]\mathbf{g(450) = 403.65 - 3.3 }[/tex]

[tex]\mathbf{g(450) = 400.35}[/tex]

Hence, you would receive 400.35 Euros, when you exchange 450 dollars

(c) Dollar equivalent of 60 Euros

In (a), we have [tex]\mathbf{g(d) = 0.897d - 3.3 }[/tex]

Substitute 60 for g(d)

[tex]\mathbf{60 = 0.897d - 3.3 }[/tex]

Add 3.3 to both sides

[tex]\mathbf{0.897d = 63.3 }[/tex]

Divide both sides by 0.897

[tex]\mathbf{d = 70.6}[/tex]

Hence, you would receive 70.6 dollars, when you exchange 60 Euros

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