find the interest rate and how long will it take for an investment to grow
Calculus

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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.
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what is the correct answer for the first question
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