you deposit money into a savings account that earns interest. the balance $b$ of the account after $t$ years…

you deposit money into a savings account that earns interest. the balance $b$ of the account after $t$ years is given by $b = 1200\times1.05^t$ dollars.\n(a) make a table that shows the account balance for years 0 through 10. (round your answers to one - decimal place.)\n(b) you want to buy a home entertainment center that costs $1500$. how long will you need to wait to have enough money in your savings account? (report your answer to the nearest whole year.)

you deposit money into a savings account that earns interest. the balance $b$ of the account after $t$ years is given by $b = 1200\times1.05^t$ dollars.\n(a) make a table that shows the account balance for years 0 through 10. (round your answers to one - decimal place.)\n(b) you want to buy a home entertainment center that costs $1500$. how long will you need to wait to have enough money in your savings account? (report your answer to the nearest whole year.)

Answer

Explanation:

Step1: Calculate balance for year 0

Substitute $t = 0$ into $B=1200\times1.05^t$. Since any non - zero number to the power of 0 is 1, $B = 1200\times1.05^0=1200\times1 = 1200$.

Step2: Calculate balance for year 1

Substitute $t = 1$ into $B = 1200\times1.05^t$. Then $B=1200\times1.05^1=1200\times1.05 = 1260.0$.

Step3: Calculate balance for year 2

Substitute $t = 2$ into $B = 1200\times1.05^t$. So $B=1200\times1.05^2=1200\times1.1025 = 1323.0$.

Step4: Calculate balance for year 3

Substitute $t = 3$ into $B = 1200\times1.05^t$. We get $B=1200\times1.05^3=1200\times1.157625 = 1389.2$.

Step5: Calculate balance for year 4

Substitute $t = 4$ into $B = 1200\times1.05^t$. Thus $B=1200\times1.05^4=1200\times1.21550625 = 1458.6$.

Step6: Calculate balance for year 5

Substitute $t = 5$ into $B = 1200\times1.05^t$. Then $B=1200\times1.05^5=1200\times1.2762815625 = 1531.5$.

Step7: Calculate balance for year 6

Substitute $t = 6$ into $B = 1200\times1.05^t$. So $B=1200\times1.05^6=1200\times1.340095640625 = 1608.1$.

Step8: Calculate balance for year 7

Substitute $t = 7$ into $B = 1200\times1.05^t$. We have $B=1200\times1.05^7=1200\times1.40710042265625 = 1688.5$.

Step9: Calculate balance for year 8

Substitute $t = 8$ into $B = 1200\times1.05^t$. Thus $B=1200\times1.05^8=1200\times1.4774554437890625 = 1772.9$.

Step10: Calculate balance for year 9

Substitute $t = 9$ into $B = 1200\times1.05^t$. Then $B=1200\times1.05^9=1200\times1.551328215978515625 = 1861.6$.

Step11: Calculate balance for year 10

Substitute $t = 10$ into $B = 1200\times1.05^t$. So $B=1200\times1.05^{10}=1200\times1.62889462677763671875 = 1954.7$.

For part (b), we set $B = 1500$ and solve the equation $1500=1200\times1.05^t$ for $t$.

Step1: Rearrange the equation

First, divide both sides of the equation $1500 = 1200\times1.05^t$ by 1200. We get $\frac{1500}{1200}=1.05^t$, which simplifies to $1.25 = 1.05^t$.

Step2: Take the natural logarithm of both sides

$\ln(1.25)=\ln(1.05^t)$. Using the property of logarithms $\ln(a^b)=b\ln(a)$, we have $\ln(1.25)=t\ln(1.05)$.

Step3: Solve for t

$t=\frac{\ln(1.25)}{\ln(1.05)}\approx\frac{0.2231}{0.0488}\approx 4.6$. Rounding up to the next whole year, $t = 5$ years.

$t$ (years) $B$ (dollars)
0 1200.0
1 1260.0
2 1323.0
3 1389.2
4 1458.6
5 1531.5
6 1608.1
7 1688.5
8 1772.9
9 1861.6
10 1954.7

Answer:

(a) See the above - made table. (b) 5 years