Redo problem 8 in section 6.3 of your textbook (page 288) assuming that the parents need $105000 in 9 years for college expenses, and that the bank account earns 9.25% compounded continuously. Round your answers to the nearest cent. (You may need to compute your answers to 4 or more decimal places before you round to the nearest cent.)
a) At what constant, continuous rate must the parents deposit money into the account in order to save the money?
Answer:
b) If the parents instead deposit a lump sum now, how much must the deposit be to attain the goal?
Answer:
To solve problem 8 in section 6.3 of your textbook, we need to use the formula for continuous compound interest:
A = P * e^(rt)
Where:
A = the final amount
P = the principal (initial deposit)
e = the base of the natural logarithm (approximately 2.71828)
r = the interest rate (in decimal form)
t = the time period (in years)
a) To find the constant continuous rate at which the parents must deposit money, we need to solve for r in the formula. We know the final amount A is $105,000, the time period t is 9 years, and the interest rate r is unknown.
105,000 = P * e^(r * 9)
To solve for r, we need to isolate it. Divide both sides of the equation by P:
e^(r * 9) = 105,000 / P
Now, take the natural logarithm (ln) of both sides to get rid of the exponential function:
ln(e^(r * 9)) = ln(105,000 / P)
Using the property of logarithms (ln(e^x) = x), we can simplify the equation:
r * 9 = ln(105,000 / P)
Now, isolate r by dividing both sides by 9:
r = ln(105,000 / P) / 9
So, the parents must deposit money at a constant, continuous rate of ln(105,000 / P) / 9 where P is the initial deposit.
b) To find the lump sum deposit necessary to attain the goal, we can rearrange the formula and solve for P:
A = P * e^(rt)
Divide both sides by e^(rt):
A / e^(rt) = P
Substitute the given values into the equation: A = $105,000, r = 0.0925 (9.25% as a decimal), and t = 9 years:
P = $105,000 / e^(0.0925 * 9)
Using a calculator or computer, calculate e^(0.0925 * 9) and divide $105,000 by the result to find the deposit amount required to attain the goal.
Round your answers to the nearest cent for both parts of the problem.