Question

Carl is saving up money for a down payment on a motorcycle. He currently has $2516, but knows he can get a loan at a lower interest rate if he can put down $3276. If he invests the $2516 in an account that earns 4.5% annually, compounded continuously, how long will it take Carl to accumulate the $3276? Round your answer to two decimal places, if necessary.

Options:
A) 1.25 years
B) 2.50 years
C) 3.75 years
D) 5.00 years

Answer 1

Carl's investment of $2516 will grow to $3276 at a 4.5% annual compounded continuous interest rate. Using the formula for continuous compounding, the time required to reach his goal can be calculated, which will then be rounded to two decimal places to determine the correct option.

Step-by-step explanation:

Carl is looking to determine how long it will take for his investment of $2516 at an annual compounded continuously interest rate of 4.5% to grow to $3276. To solve this, we use the formula for continuous compounding, which is A = Pert, where A is the future value of the investment, P is the principal amount, r is the interest rate, and t is the time in years the money is invested for.

To find t, we rearrange the formula to t = ln(A/P) / r. Plugging in the given numbers, we get t = ln(3276/2516) / 0.045, which can be calculated to give the time needed for the investment to reach the desired amount.

After performing the calculation, the answer which should be rounded to two decimal places to fit the options given would determine the time Carl needs to wait before he has enough for the down payment.