David invested $68,000 in an account paying an interest rate of 5.5% compounded daily. Assuming no deposits or withdrawals are made, how long would it take, to the nearest tenth of a year, for the value of the account to reach $93,000?
To find out how long it would take for the value of the account to reach $93,000, you will need to use the formula for compound interest.
The formula for compound interest is given by:
A = P(1 + r/n)^(nt)
Where:
A = the future value of the investment/loan, which is $93,000 in this case
P = the principal amount, which is $68,000
r = the annual interest rate, which is 5.5% (convert to decimal by dividing by 100, so r = 0.055)
n = the number of times that interest is compounded per year, which is daily, so n = 365 (since there are 365 days in a year)
t = the number of years
We need to find the value of t (time). Rearranging the formula, we have:
t = (log(A/P)) / (n * log(1 + r/n))
Now, we can substitute the given values into the formula and calculate the time it would take for the account to reach $93,000.
t = (log(93000/68000)) / (365 * log(1 + 0.055/365))
Using a calculator, evaluate the expression to find t.
t ≈ 7.2 years
Therefore, it would take approximately 7.2 years (to the nearest tenth of a year) for the value of the account to reach $93,000.
To solve this problem, we can use the formula for compound interest:
A = P(1 + r/n)^(nt)
Where:
A = the future value of the investment (in this case, $93,000)
P = the principal amount (in this case, $68,000)
r = the annual interest rate (in this case, 5.5% or 0.055)
n = the number of times interest is compounded per year (in this case, daily, so n = 365)
t = the time in years (which we need to find)
Substituting the given values into the formula, we get:
93,000 = 68,000(1 + 0.055/365)^(365t)
To find the value of t, we need to isolate it. We can start by dividing both sides of the equation by 68,000:
93,000/68,000 = (1 + 0.055/365)^(365t)
Simplifying the left side gives:
1.3676470588235294 = (1 + 0.055/365)^(365t)
Now, we can take the natural logarithm (ln) of both sides to remove the exponent:
ln(1.3676470588235294) = ln[(1 + 0.055/365)^(365t)]
Using the property of logarithms that ln(a^b) = b * ln(a), we get:
ln(1.3676470588235294) = 365t * ln(1 + 0.055/365)
Next, we can divide both sides of the equation by 365 * ln(1 + 0.055/365):
ln(1.3676470588235294) / (365 * ln(1 + 0.055/365)) = t
Calculating the left side of the equation gives:
t ≈ 0.1565
Therefore, it would take approximately 0.1565 years or 1.9 months (to the nearest tenth of a year) for the value of the account to reach $93,000.