A savings account is started with an initial deposit of $500. The account earns 1.5% interest compounded annually.
Write an equation to represent the amount of money in the account as a function of time in years.
Find the amount of time it takes for the account balance to reach $800. Show your work.
To represent the amount of money in the account as a function of time in years, we can use the formula for compound interest:
A = P(1 + r/n)^(nt)
Where:
A = the final amount (account balance)
P = the initial deposit
r = the interest rate (in decimal form)
n = the number of times interest is compounded per year
t = the time in years
In this case, the initial deposit (P) is $500, the interest rate (r) is 1.5% (or 0.015), and the account balance (A) is the function we want to find.
Therefore, the equation representing the amount of money in the account as a function of time is:
A = 500(1 + 0.015/1)^(1t)
Simplifying further,
A = 500(1.015)^t
Now, to find the amount of time it takes for the account balance to reach $800, we need to solve the equation:
800 = 500(1.015)^t
Divide both sides of the equation by 500:
1.6 = (1.015)^t
To solve for t, we can take the natural logarithm of both sides:
ln(1.6) = ln((1.015)^t)
Using the properties of logarithms, we can bring the exponent down:
ln(1.6) = t ln(1.015)
Divide both sides by ln(1.015):
t = ln(1.6) / ln(1.015)
Using a calculator, we find:
t ≈ 52.6717 years
Therefore, it takes approximately 52.6717 years for the account balance to reach $800.
To write an equation that represents the amount of money in the account as a function of time in years, you can use the compound interest formula:
A = P(1 + r/n)^(nt)
Where:
A = the amount of money in the account after time t
P = the initial deposit
r = the annual interest rate (expressed as a decimal)
n = the number of times interest is compounded per year
t = time in years
In this case, the initial deposit is $500, the annual interest rate is 1.5% (or 0.015 as a decimal), and interest is compounded annually. Therefore, n = 1.
The equation for the amount of money in the account as a function of time becomes:
A = 500(1 + 0.015/1)^(1t)
A = 500(1.015)^t
Now, to find the amount of time it takes for the account balance to reach $800, we can set up the equation:
800 = 500(1.015)^t
To solve for t, we can divide both sides of the equation by 500:
1.6 = (1.015)^t
Now, to solve for t, we can take the natural logarithm (ln) of both sides of the equation:
ln(1.6) = ln[(1.015)^t]
Using the property of logarithms which states that ln(a^b) = b * ln(a), the equation simplifies to:
ln(1.6) = t * ln(1.015)
Finally, we can solve for t by dividing both sides of the equation by ln(1.015):
t = ln(1.6) / ln(1.015)
Using a calculator, we find that t is approximately 2.397 years.
Therefore, it takes approximately 2.397 years for the account balance to reach $800.