A spy is trying to escape. She must get by four guards. Each guard takes half of her remaining money. She then $64 for her plane ticket. How much money does she have to start with in order to make her escape?

Accepted Solution

Answer:The spy must have at the start the amount of $1,024 in order to escapeStep-by-step explanation:Leta ------> amount of money that the spy must have at the start to escapey ----> the remaining money x ----> the number of guardsIn 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]wherea is the initial value (amount of money at the start)b is the baseb=(1-r)r is the rate of decayIn this problem we haver=50% -----> r=0.50The value of b isb=(1-0.50)=0.50substitute[tex]y=a(0.50^{x})[/tex]we know thatIn order to escape after the fourth guard the amount of money remaining must be equal to $64soFor x=4, y=$64substitute 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]thereforeThe spy must have at the start the amount of $1,024 in order to escape