f $20,000 is invested in a savings account offering 3.5% per year, compounded continuously, how fast is the balance growing after 4 years? (Round your answer to the nearest cent.)

$ 1 per yr

The effective interest rate with continuous componding is

i = e^(.035)-1 = 3.562%

After four years the balance is
(with continuous compounding)
20,000*(1.0362)^4 = $23,005.48.

The balance is growing at an instantaneous rate of
0.035*23,005.48 = $805.19 /year

but over the year to follow, the earnings will be
$832.80 /yr

Sorry, how did you get the 1.0362?

thanks

Well, buckle up because we're taking a wild ride on the compounding train!

To calculate the growth rate, we can use the continuous compounding formula:

A = P * e^(rt)

Where:
A = final balance
P = initial investment
e = mathematical constant approximately equal to 2.71828
r = annual interest rate (as a decimal)
t = time in years

Plugging in the values:

A = 20000 * e^(0.035 * 4)

So, let's grab our math clown shoes and calculate:

A ≈ 20000 * 2.71828^(0.14)

After doing the math magic, we find that:

A ≈ 20000 * 1.150268

Drumroll, please...

A ≈ $23,005.36

Hold on to your hat! The balance is growing at an approximate rate of $1 per year. So, after 4 years, the balance would have increased by a total of $4.

Remember though, this is just an approximation, so don't go planning your retirement around my clown calculations!

To find out how fast the balance is growing after 4 years, we need to use the formula for continuous compound interest:

A = P * e^(rt)

Where:
A = the final balance
P = the initial principal (amount invested)
e = the mathematical constant approximately equal to 2.71828
r = the annual interest rate (in decimal form)
t = the time in years

In this case:
P = $20,000
r = 3.5% = 0.035 (converted to decimal form)
t = 4 years

Plugging in the values into the formula, we get:

A = 20000 * e^(0.035 * 4)

Now we need to find the derivative of A with respect to time (t) to determine how fast the balance is growing.

dA/dt = 20000 * 0.035 * e^(0.035 * 4)

Evaluating this expression will give us the rate at which the balance is growing after 4 years.

Please note that this is a continuous compound interest problem, so the growth rate is continuously changing.