SOLUTION: An initial deposit is made in a bank account. Find the interest rate, r , if the interest is compounded continuously and no withdrawals or further deposits are made. Round to the n

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Question 1139828: An initial deposit is made in a bank account. Find the interest rate, r , if the interest is compounded continuously and no withdrawals or further deposits are made. Round to the nearest hundredth of a percent.
Initial Amount: $3,500
Amount in 5 years: $5,100
Found 2 solutions by jim_thompson5910, MathTherapy:
Answer by jim_thompson5910(35256) (Show Source):
You can put this solution on YOUR website!

Answer: 7.53%

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How I got this answer:

The formula to use is which computes the value of an account when interest is compounded continuously
A = final amount in account after t years
P = initial deposit (this is a one time deposit; money can't be added or taken out during the t years)
r = annual interest rate in decimal form
t = number of years
side note: you can make t be something like months or days or whatever time unit you want, but keep in mind that r must be adjusted as well.

The 'e' refers to e = 2.718... which is similar to pi = 3.14... in that it's a constant where the decimal digits go on forever without any known pattern. The constant 'e' is often used for any kind of exponential growth, which is effectively what continuously compounded interest problems deal with.

In this case,
A = 5100 is the amount we want after five years
P = 3500 is the initial amount
r = unknown (what we want to solve for)
t = 5 is the number of years that elapse

We will use logarithms to help isolate r. Specifically we'll use a natural logarithm which is abbreviated as "ln", which in uppercase format would look like "LN" or you can just make the L uppercase so that you don't confuse it with the letter 'i'.

Plug in the given info

Divide both sides by 3500

Note how the two "3500" values on the right hand side cancel out

Now the "3500" terms are gone completely on the right hand side

Divide 5100/3500 to get approximately 1.45714285714286

Swap both sides

Apply the natural log Ln to both sides

Use the rule that Ln(x^y) = y*Ln(x) to pull down the exponent

Use the rule that Ln(e) = 1

Simplify

Use a calculator to compute Ln(1.45714285714286) to get roughly 0.37647757123491

Divide both sides by 5

The "5"s on the left hand side cancel out

The "5"s on the left hand side are gone now. The variable r is fully isolated.

Use a calculator to compute 0.37647757123491/5 to get roughly 0.07529551424699

The last step is to multiply this value by 100 to convert to percentage form
100*0.07529551424699 = 7.529551424699%
This rounds to 7.53% which is the final answer

Side note: my steps may be a bit too wordy and drawn out; however, I did so to help show all of the algebra involved just in case you may get confused by any step. The work you'll show to your teacher will most likely be more condensed and quicker to the point.

Answer by MathTherapy(10552) (Show Source):
You can put this solution on YOUR website!
An initial deposit is made in a bank account. Find the interest rate, r , if the interest is compounded continuously and no withdrawals or further deposits are made. Round to the nearest hundredth of a percent.
Initial Amount: $3,500
Amount in 5 years: $5,100

Formula for interest rate (r), with future value (FV), present value (PV), and time (t), in years, KNOWN:

OR
Using the future value formula for continuous compounding, or, , we get:
------- Substituting 5,100 for A, 3,500 for P, and 5 for r

-------- Converting to EXPONENTIAL form