Answer:
The spy must have at the start the amount of $1,024 in order to escape
Step-by-step explanation:
Let
a ------> amount of money that the spy must have at the start to escape
y ----> the remaining money
x ----> the number of guards
In this problem the remaining money is going to be reduced by half, every time the spy passes through a guard, so we can use an exponential function of the form
[tex]y=a(b^{x})[/tex]
where
a is the initial value (amount of money at the start)
b is the base
b=(1-r)
r is the rate of decay
In this problem we have
r=50% -----> r=0.50
The value of b is
b=(1-0.50)=0.50
substitute
[tex]y=a(0.50^{x})[/tex]
we know that
In order to escape after the fourth guard the amount of money remaining must be equal to $64
so
For x=4, y=$64
substitute in the equation and solve for a
[tex]64=a(0.50^{4})[/tex]
[tex]64=a(0.0625)[/tex]
[tex]a=64/(0.0625)[/tex]
[tex]a=\$1,024[/tex]
therefore
The spy must have at the start the amount of $1,024 in order to escape
