Complete the table to determine the balance A for $20,000 invested at rate r = 6% for t years, compounded continuously.

chart looks like this:
t 10 20 30 40 50
A ___ ___ ___ ___ ____
I need to answer A

I will do the first one for t=10

you do the others the same way

in general: A = 20000 e^(rt) , were r is the rate in decimals

if t = 10
A = 20000(e^(.06(10))
= 36442.38

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Oh, you want me to fill in the blanks? Alright, here we go.

t 10 20 30 40 50
A $35,690.51 $79,220.74 $176,213.94 $391,749.18 $869,779.70

Wow, that investment is really compounding like crazy! It's like watching a clown juggling balls - the numbers just keep getting bigger and bigger! Don't expect me to juggle real balls though, I'm more of a pun-loving bot.

To determine the balance, A, for $20,000 invested at a rate of 6% for t years, compounded continuously, we can use the formula for continuous compound interest:

A = P * e^(rt)

Where:
A = balance (final amount)
P = principal (initial investment)
r = interest rate (expressed as a decimal)
t = time (in years)
e = Euler's number (approximately 2.71828)

Let's calculate the values for A using the given table:

For t = 10 years:
A = 20000 * e^(0.06 * 10)
A ≈ 20000 * e^0.6
A ≈ 20000 * 1.82212
A ≈ 36442.40

For t = 20 years:
A = 20000 * e^(0.06 * 20)
A ≈ 20000 * e^1.2
A ≈ 20000 * 3.32012
A ≈ 66402.40

For t = 30 years:
A = 20000 * e^(0.06 * 30)
A ≈ 20000 * e^1.8
A ≈ 20000 * 6.04965
A ≈ 120993.00

For t = 40 years:
A = 20000 * e^(0.06 * 40)
A ≈ 20000 * e^2.4
A ≈ 20000 * 11.02318
A ≈ 220463.60

For t = 50 years:
A = 20000 * e^(0.06 * 50)
A ≈ 20000 * e^3.0
A ≈ 20000 * 20.08554
A ≈ 401710.80

The completed table is as follows:

t | 10 | 20 | 30 | 40 | 50
A | 36442.40 | 66402.40 | 120993.00 | 220463.60 | 401710.80

To determine the balance A for $20,000 invested at a continuously compounded interest rate of 6% for different periods of time t, you can use the formula:

A = P*e^(rt)

Where:
A is the balance or amount after time t
P is the principal amount (initial investment) which is $20,000 in this case
e is Euler's number, approximately equal to 2.71828
r is the interest rate expressed as a decimal (6% as 0.06)
t is the time in years

Now, let's calculate the balance A for each given value of t in the table:

For t = 10 years:
A = 20000 * e^(0.06 * 10)
A ≈ 20000 * e^0.6
A ≈ 20000 * 1.82212
A ≈ 36442.4

For t = 20 years:
A = 20000 * e^(0.06 * 20)
A ≈ 20000 * e^1.2
A ≈ 20000 * 3.32012
A ≈ 66402.4

For t = 30 years:
A = 20000 * e^(0.06 * 30)
A ≈ 20000 * e^1.8
A ≈ 20000 * 6.04965
A ≈ 120993

For t = 40 years:
A = 20000 * e^(0.06 * 40)
A ≈ 20000 * e^2.4
A ≈ 20000 * 11.02318
A ≈ 220464

For t = 50 years:
A = 20000 * e^(0.06 * 50)
A ≈ 20000 * e^3
A ≈ 20000 * 20.0855
A ≈ 401710

So, the completed table would look like this:

t 10 20 30 40 50
A 36442.4 66402.4 120993 220464 401710