Answer: day six. A=3, b=2, t=0 to t=5, covering 6 days. $3, $6, $12, $24, $48, $96.
Step-by-step explanation:
. A is the starting amount at t=0. b is the base, two because the amount doubles each day. b would be 3 if it tripled every day. t is the final day number, counting from zero.
![A=3,b=2,t=0,Ab^t=3\times2^0=3](https://tex.z-dn.net/?f=A%3D3%2Cb%3D2%2Ct%3D0%2CAb%5Et%3D3%5Ctimes2%5E0%3D3)
![A=3,b=2,t=1,Ab^t=3\times2^1=6](https://tex.z-dn.net/?f=A%3D3%2Cb%3D2%2Ct%3D1%2CAb%5Et%3D3%5Ctimes2%5E1%3D6)
![A=3,b=2,t=2,Ab^t=3\times2^2=12](https://tex.z-dn.net/?f=A%3D3%2Cb%3D2%2Ct%3D2%2CAb%5Et%3D3%5Ctimes2%5E2%3D12)
![A=3,b=2,t=3,Ab^t=3\times2^3=24](https://tex.z-dn.net/?f=A%3D3%2Cb%3D2%2Ct%3D3%2CAb%5Et%3D3%5Ctimes2%5E3%3D24)
![A=3,b=2,t=4,Ab^t=3\times2^4=48](https://tex.z-dn.net/?f=A%3D3%2Cb%3D2%2Ct%3D4%2CAb%5Et%3D3%5Ctimes2%5E4%3D48)
![A=3,b=2,t=5,Ab^t=3\times2^0=96](https://tex.z-dn.net/?f=A%3D3%2Cb%3D2%2Ct%3D5%2CAb%5Et%3D3%5Ctimes2%5E0%3D96)
We can also do this with units included.
Obviously the unit for A is dollars, and t is in days. But what is the unit for b?
Well, the amount doubles once _per day_. So try b = 2 per day = 2/day. That doesn't work. The correct value for b including units is
.
So if you wanted to express time in weeks while still doubling once per day, you would multiply by a value equivalent to one,
, and cancel units:
.
![A=\$3, b=2^\left(\frac{1}{\text{day}}\right), t=5\,\text{day},\\Ab^t=\$3\big\left(2^\left(\frac{1}{\text{day}}\right)\big\right)^{5\,\text{day}}=\$3\,2^5=\$96](https://tex.z-dn.net/?f=A%3D%5C%243%2C%20b%3D2%5E%5Cleft%28%5Cfrac%7B1%7D%7B%5Ctext%7Bday%7D%7D%5Cright%29%2C%20t%3D5%5C%2C%5Ctext%7Bday%7D%2C%5C%5CAb%5Et%3D%5C%243%5Cbig%5Cleft%282%5E%5Cleft%28%5Cfrac%7B1%7D%7B%5Ctext%7Bday%7D%7D%5Cright%29%5Cbig%5Cright%29%5E%7B5%5C%2C%5Ctext%7Bday%7D%7D%3D%5C%243%5C%2C2%5E5%3D%5C%2496)