Answer:
1. 10.7 years
2. 17.0 years
3. 2nd option
Step-by-step explanation:
Use formula for compounded interest
![A=P\cdot \left(1+\dfrac{r}{n}\right)^{nt},](https://tex.z-dn.net/?f=A%3DP%5Ccdot%20%5Cleft%281%2B%5Cdfrac%7Br%7D%7Bn%7D%5Cright%29%5E%7Bnt%7D%2C)
where
A is final value, P is initial value, r is interest rate (as decimal), n is number of periods and t is number of years.
In your case,
1. P=$5745, A=2P=$11490, n=12 (compounded monthly), r=0.065 (6.5%) and t is unknown. Then
![11490=5745\cdot \left(1+\dfrac{0.065}{12}\right)^{12t},\\ \\2=(1.0054)^{12t},\\ \\12t=\log_{1.0054}2,\\ \\t=\dfrac{1}{12}\log_ {1.0054}2\approx 10.7\ years.](https://tex.z-dn.net/?f=11490%3D5745%5Ccdot%20%5Cleft%281%2B%5Cdfrac%7B0.065%7D%7B12%7D%5Cright%29%5E%7B12t%7D%2C%5C%5C%20%5C%5C2%3D%281.0054%29%5E%7B12t%7D%2C%5C%5C%20%5C%5C12t%3D%5Clog_%7B1.0054%7D2%2C%5C%5C%20%5C%5Ct%3D%5Cdfrac%7B1%7D%7B12%7D%5Clog_%20%7B1.0054%7D2%5Capprox%2010.7%5C%20years.)
2. P=$5745, A=3P=$17235, n=12 (compounded monthly), r=0.065 (6.5%) and t is unknown. Then
![3=(1.0054)^{12t},](https://tex.z-dn.net/?f=3%3D%281.0054%29%5E%7B12t%7D%2C)
![12t=\log_{1.0054}3,](https://tex.z-dn.net/?f=12t%3D%5Clog_%7B1.0054%7D3%2C)
![t=\dfrac{1}{12}\log_{1.0054}3\approx 17.0\ years.](https://tex.z-dn.net/?f=t%3D%5Cdfrac%7B1%7D%7B12%7D%5Clog_%7B1.0054%7D3%5Capprox%2017.0%5C%20years.)
3. <u>1 choice:</u> P=$5745, n=12 (compounded monthly), r=0.065 (6.5%), t=5 years and A is unknown. Then
![A=5745\cdot \left(1+\dfrac{0.065}{12}\right)^{12\cdot 5},\\ \\A=5745\cdot (1.0054)^{60}\approx \$7936.39.](https://tex.z-dn.net/?f=A%3D5745%5Ccdot%20%5Cleft%281%2B%5Cdfrac%7B0.065%7D%7B12%7D%5Cright%29%5E%7B12%5Ccdot%205%7D%2C%5C%5C%20%5C%5CA%3D5745%5Ccdot%20%281.0054%29%5E%7B60%7D%5Capprox%20%5C%247936.39.)
<u>2 choice:</u> P=$5745, n=4 (compounded monthly), r=0.0675 (6.75%), t=5 years and A is unknown. Then
![A=5745\cdot \left(1+\dfrac{0.0675}{4}\right)^{4\cdot 5},\\ \\A=5745\cdot (1.0169)^{20}\approx \$8032.58.](https://tex.z-dn.net/?f=A%3D5745%5Ccdot%20%5Cleft%281%2B%5Cdfrac%7B0.0675%7D%7B4%7D%5Cright%29%5E%7B4%5Ccdot%205%7D%2C%5C%5C%20%5C%5CA%3D5745%5Ccdot%20%281.0169%29%5E%7B20%7D%5Capprox%20%5C%248032.58.)
The best will be 2nd option, because $8032.58>$7936.39