Patty's rate is a constant, $10 per hour.
Tyler's rate is exponential, or it can be said if forms a geometric sequence, having the form:
f=ir^(t-1), f=final value, i=initial value, r=common ratio or "rate", t=time
We can see that the common ratio is 2 because 2/1=4/2=2 is constant. Each term is twice the previous term. And we also know that the initial value is $1 so:
f=1(2^(t-1)) or just
T=2^(t-1)
We want to know when T>P so
2^(t-1)>10 taking the natural log of both sides
(t-1)ln2>ln10 dividing both sides by ln2
t-1>ln10/ln2 add 1 to both sides
t>(ln10/ln2)+1
t>4.32 (to the nearest hundredth of an hour)
Since I guess that we should assume that the rates only change after incremental or integer values for t.
Tyler will earn more per hour than Patty starting in the 5th hour.
P(5)=$10 per hour
T(5)=2^(5-1)=2^4=$16 per hour
