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The first part of the question would be using the equation: A = Pe^rt, because it states that the interest is compounded continuously. A is the total amount, P is the principal amount, e represents that it is an exponential function, r is the interest rate (as a decimal), and t is the time in years.
So we would plug what we know into the equation:
A = Pe^rt
7000 = 4000e^r(10)
7000 = 4000e^10r ; Divide both sides by 4000
1.75 = e^10r ; now take the natural logs of both sides (it will look like “In” on your calculator)
In(1.75) = In(e^10r)
In logarithms, there is a rule stating that log(subscript b)B^ A = A; we would use this rule to eliminate In(e^10r) to 10r.
Therefore: In(1.75) = 10r ; now we would divide each side by 10.
In(1.75) / 10 = r
0.055961578 = r when rounded we get 0.06 = r. We change this to a percent, r = 6% (to change to percent, you take 0.06 x 100).
Now for the second question:
How long will it take for an investment of $14,000 to triple if the investment earns interest at the rate of 5%/year compounded continuously?
For $14,000 to triple, the final value must be $42,000. We will be using the same equation A = Pe^rt and will be solving for t (time in years).
A = Pe^rt
42,000 = 14, 000e^0.05t; we will divide both sides by 14,000
3 = e^0.05t; we will then take the natural logs of both sides (In)
In(3) = In(e^0.05t); we will again use the same logarithm rule used in the previous question so that In(e^0.05t) will be simplified as 0.05t.
In(3) = 0.05t; we will divide each side by 0.05 to isolate for t
In(3) / 0.05 = t
21.97224577 = t, rounded it will be 22 years.
Therefore, it will take around 22 years for the initial value of $14,000 to triple.
Thank you for the opportunity for me to help you out! Please let me know if you need any clarification. I'm always happy to answer your questions. Take care~
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