If you deposit $10,000 in a savings account now, what interest rate compounded continuously would be required for you to withdraw $15,000 at the end of 6 years?
B. A savings and loans facility offers a CD with a monthly compounding rate that has an APY of 6.25%
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To find the interest rate required for your savings account, compounded continuously, in order to withdraw $15,000 at the end of 6 years, you can use the formula for continuous compounding:
A = P * e^(rt)
Where:
A is the final amount (in this case, $15,000)
P is the principal (initial deposit, in this case, $10,000)
e is the base of the natural logarithm (approximately 2.71828)
r is the interest rate (what we're trying to find)
t is the time in years (in this case, 6 years)
Rearranging the formula, we can solve for the interest rate (r):
r = ln(A/P) / t
Substituting the given values:
r = ln(15000/10000) / 6
Using a calculator:
r ≈ ln(1.5) / 6
r ≈ 0.405 / 6
r ≈ 0.0675
Therefore, the interest rate required for your savings account, compounded continuously, to withdraw $15,000 at the end of 6 years is approximately 6.75%.
Moving on to the second question, if a savings and loans facility offers a CD with a monthly compounding rate that has an APY (Annual Percentage Yield) of 6.25%, it means that the interest is compounded monthly and the stated APY takes into account the effect of compounding over the course of a year.
To determine the monthly interest rate, divide the APY by 12 (the number of months in a year):
Monthly Interest Rate = APY / 12
In this case:
Monthly Interest Rate = 6.25% / 12
Using a calculator:
Monthly Interest Rate = 0.0625 / 12
Monthly Interest Rate ≈ 0.00521
Therefore, the monthly interest rate for the CD with an APY of 6.25% is approximately 0.00521, or 0.521%.