suppose you won a contest at the start of 5th grade that deposited $3000 in an account that pays 5% annual interest compounded continuously. how much will you have in the account when you enter high school 4 years later? round to nearest dollar how much will be in the account when you graduate from high school?
compounded continuously you have to use e^rt
Pe^rt
3000(e)^[(.05)(4)]
Well, well, well! A contest winner, are we? Congratulations! Now let's crunch those numbers and have some fun!
To calculate the amount of money you'll have in the account when you enter high school, we can use the formula for continuous compound interest:
A = P * e^(rt)
Where:
A = Final amount
P = Principal amount (initial deposit)
e = Euler's number (approximately 2.71828)
r = Interest rate (in decimal form)
t = Time (in years)
So, let's plug in the values and see what we get:
P = $3000
r = 5% = 0.05
t = 4 years
A = 3000 * e^(0.05 * 4)
Now, let's whip out our trusty calculator and find out the answer! *drum roll*
After four years, you'll have approximately $3435 in the account when you enter high school! Nice!
But wait, there's more! Let's move on to when you graduate from high school. Assuming you spend four years in high school, we can apply the same formula, but this time with a time of 8 years:
A = 3000 * e^(0.05 * 8)
And the grand total in your account when you graduate high school is approximately $3976!
Well, isn't that dandy? You've grown your money without even lifting a finger. Time to celebrate! 🎉
To calculate the amount you will have in the account when you enter high school 4 years later, we can use the formula for continuously compounded interest:
A = P * e^(rt),
where:
A = the amount in the account after time t
P = the initial principal (in this case, $3000)
e = the mathematical constant approximately equal to 2.71828
r = the annual interest rate (in decimal form, 5% = 0.05)
t = the time in years (4 years)
To calculate the amount, we substitute the values into the formula:
A = $3000 * e^(0.05 * 4).
Using a scientific calculator or an online calculator, we can find:
A ≈ $3000 * e^(0.2) ≈ $3000 * 1.221402758 ≈ $3664.209274.
Therefore, you will have approximately $3664 in the account when you enter high school.
To calculate the amount in the account when you graduate from high school, we'll need to calculate the total time in years. Assuming you complete high school in four years, the total time will be 8 years.
Using the same formula:
A = P * e^(rt),
where:
A = the amount in the account after time t
P = the initial principal ($3000)
e = the mathematical constant approximately equal to 2.71828
r = the annual interest rate (0.05)
t = the time in years (8 years)
A = $3000 * e^(0.05 * 8).
Using a calculator again, we can find:
A ≈ $3000 * e^(0.4) ≈ $3000 * 1.491824697 ≈ $4475.47409.
Therefore, you will have approximately $4475 in the account when you graduate from high school.
To calculate the amount in the account when you enter high school and graduate from high school, we need to use the formula for continuous compound interest:
A = P * e^(rt)
Where:
A = the final amount in the account
P = the initial principal (the amount deposited in the account)
e = Euler's number, approximately 2.71828
r = the interest rate (in decimal form)
t = the time (in years)
Let's calculate the amount in the account when you enter high school (4 years later):
P = $3000 (initial amount)
r = 0.05 (5% annual interest rate, expressed as a decimal)
t = 4 years
Plugging these values into the formula:
A = 3000 * e^(0.05 * 4)
Now, let's solve it step by step:
Step 1: Calculate the value inside the exponent: 0.05 * 4 = 0.2
Step 2: Compute e^0.2 using the constant e (approximately 2.71828). This gives us 1.22140 (rounded to 5 decimal places).
Step 3: Multiply the initial principal by the value from step 2:
A = 3000 * 1.22140
A ≈ $3,664 (rounded to the nearest dollar)
Therefore, when you enter high school 4 years later, the amount in the account will be approximately $3,664.
Now, let's calculate the amount in the account when you graduate from high school, which is after an additional 4 years.
P = $3,664 (the amount when you entered high school)
r = 0.05 (5% annual interest rate, expressed as a decimal)
t = 4 years
Using the same formula as before:
A = 3664 * e^(0.05 * 4)
We can repeat the steps to find the amount in the account:
Step 1: Calculate the value inside the exponent: 0.05 * 4 = 0.2
Step 2: Compute e^0.2: approximately 1.22140 (rounded to 5 decimal places)
Step 3: Multiply the initial principal by the value from step 2:
A = 3664 * 1.22140
A ≈ $4,473 (rounded to the nearest dollar)
Therefore, when you graduate from high school, the amount in the account will be approximately $4,473.