Dee invested $3000 in a savings account that earns 6.5% interest, compounded continuously. After 4 years, he wants to buy a used car for $4000. How much money will he need to pay in addition to what is in his savings account?
We can use the formula for continuous compounding to find the balance in Dee's savings account after 4 years:
A = Pe^(rt)
where A is the balance, P is the principal (initial investment), e is the mathematical constant approximately equal to 2.71828, r is the annual interest rate as a decimal, and t is the time in years.
Plugging in the values given, we get:
A = 3000*e^(0.065*4)
A = 3000*e^0.26
A = 3000*1.297
A = 3891.60
So Dee's savings account will have a balance of $3891.60 after 4 years.
To buy the used car for $4000, Dee will need to pay an additional:
4000 - 3891.60 = $108.40
Therefore, Dee will need to pay an additional $108.40 in addition to what is in his savings account to buy the used car.
To calculate the total amount of money Dee will need to pay in addition to what is in his savings account, we need to find the amount accumulated in his savings account after 4 years with continuous compounding.
The formula to calculate the final amount with continuous compounding is given by:
A = P * e^(rt),
Where:
A is the final amount,
P is the principal amount (initial investment),
e is Euler's number (approximately 2.71828),
r is the interest rate (expressed as a decimal),
t is the time period in years.
Let's calculate the final amount in Dee's savings account:
P = $3000 (initial investment)
r = 6.5% = 0.065 (converted to decimal)
t = 4 years
Plugging these values into the formula:
A = $3000 * e^(0.065 * 4)
Now, we can calculate the final amount:
A = $3000 * e^(0.26)
A ≈ $3000 * 1.29734
A ≈ $3892.02 (rounded to two decimal places)
After 4 years, Dee will have approximately $3892.02 in his savings account. Since Dee wants to buy a used car for $4000, he will need to pay an additional:
$4000 - $3892.02 ≈ $107.98
Dee will need to pay approximately $107.98 in addition to what is in his savings account.
To determine the additional money Dee will need to pay for the used car after 4 years, we need to calculate the amount in his savings account after the given time period. We can use the formula for continuous compound interest:
A = P * e^(rt)
Where:
A = the final amount
P = the initial principal (or the amount Dee invested)
e = the mathematical constant approximately equal to 2.71828
r = the interest rate per time period (per year in this case)
t = the number of time periods (in years)
Given:
P = $3000
r = 6.5% = 0.065
t = 4 years
Substituting the values into the formula, we have:
A = 3000 * e^(0.065 * 4)
Calculating this using a calculator or software, we find:
A ≈ 3000 * e^(0.26)
A ≈ 3000 * 1.29728
A ≈ $3891.85 (rounded to two decimal places)
Therefore, after 4 years, Dee will have approximately $3891.85 in his savings account. To find out how much additional money he will need to pay for the used car, we subtract this amount from the cost of the car:
Additional money = Cost of the car - Amount in savings account
Additional money = $4000 - $3891.85
Additional money ≈ $108.15 (rounded to two decimal places)
Therefore, Dee will need to pay approximately $108.15 in addition to the amount in his savings account to buy the used car.