Respuesta :
Answer:
82 years and 3 quarters
Step-by-step explanation:
A = P (1 + r / n)ⁿˣ
P = Principal amount
r = Annual interest rate
n = Number of compounds per year
x = time in years
A = Amount after time 'x'
10500 = 2100 (1 + 0.0195 / 4)⁴ˣ
Divide the whole equation by 2100
10500 / 2100 = ({2100 (1 + 0.004875)} / 2100 )⁴ˣ
5 = (1.004875)⁴ˣ
Taking Natural logarithm (㏑) on both sides
㏑ 5 = ㏑ (1.004875)⁴ˣ
㏑ 5 = 4x ㏑ (1.004875)
1.6094 = 4x (0.004863)
1.6094 = 0.01945x
x = 82.75
So, If compounded quarterly at an APR of 1.95% the amount deposited in savings account of $2100 will accumulate to $10500 in 82 years and 3 quarters.
Using compound interest, it is found that it will take 82.7 years for the account to earn $10500.
Compound interest:
[tex]A(t) = P(1 + \frac{r}{n})^{nt}[/tex]
- A(t) is the amount of money after t years.
- P is the principal(the initial sum of money).
- r is the interest rate(as a decimal value).
- n is the number of times that interest is compounded per year.
- t is the time in years for which the money is invested or borrowed.
In this problem:
- $2100 invested, thus [tex]P = 2100[/tex].
- APR of 1.95%, thus [tex]r = 0.0195[/tex].
- Compounded quarterly, thus [tex]n = 4[/tex].
- Earn $10500, thus t for which [tex]A(t) = 10500[/tex].
Then
[tex]A(t) = P\left(1 + \frac{r}{n}\right)^{nt}[/tex]
[tex]10500 = 2100\left(1 + \frac{0.0195}{4}\right)^{4t}[/tex]
[tex](1.004875 )^{4t} = \frac{10500}{2100}[/tex]
[tex](1.004875 )^{4t} = 5[/tex]
[tex]\log{(1.004875 )^{4t}} = \log{5}[/tex]
[tex]4t\log{1.004875} = \log{5}[/tex]
[tex]t = \frac{\log{5}}{4\log{1.004875}}[/tex]
[tex]t = 82.7[/tex]
It will take 82.7 years for the account to earn $10500.
A similar problem is given at https://brainly.com/question/16051133
