The future value of an investment after t years is given by F ( t ) = 120(1.126

Question

The future value of an investment after t years is given by

F(t) = 120(1.126t) thousand dollars.

(a) Calculate the future value after 10 years. (Round your answer to three decimal places.)
$ ____ thousand

Calculate the rate of change of the future value after 10 years. (Round your answer to three decimal places.)
$ _____ thousand per year

(b) Write the linearization of F after 10 years. (Round all numerical values to three decimal places.)

FL(t) =_____

thousand dollars

(c) Use the linearization to estimate the future value after 10.5 years. (Round your answer to three decimal places.)
$ ________thousand

Explanation / Answer

Now, i assume you made a mistake in writing F(t) = 120(1.126t). I assume that it is exponential....

F(t) = 120*(1.126)^t

(a) Calculate the future value after 10 years. (Round your answer to three decimal places.)
$ ____ thousand

Solution :

Simply plug in t = 10 and find it using calculator

F(10) = 120 * (1.126)^10

F(10) = 393.1562202796013219

So, F(10) = 393.156 thousand ----> ANSWER

Calculate the rate of change of the future value after 10 years. (Round your answer to three decimal places.)
$ _____ thousand per year

We must derive the given function first

F(t) = 120(1.126^t)

So, F'(t) = 120(1.126^t)*ln(1.126)

Now, into that, plug in t = 10 :

F'(10) = 120*(1.126^10) * ln(1.126)

F'(10) = 46.656 thousand per year -----> ANSWER

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b) Write the linearization of F after 10 years. (Round all numerical values to three decimal places.)

Solution :

We found F'(t) = 120(1.126^t)*ln(1.126)

The linearization formula is :

FL(t) - F(a) = F'(a)*(t - a)

Here a = 10 for the 10 years

FL(t) - F(10) = F'(10)*(t - 10)

Plug n F(10) and F'(10) from part A above, we get :

FL(t) - 393.156 = 46.656(t - 10)

FL(t) = 46.656t - 466.56 + 393.156

FL(t) = 46.656t -73.404 ----> ANSWER

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(c) Use the linearization to estimate the future value after 10.5 years. (Round your answer to three decimal places.)

Take part B's equation and simply plug in t with 10.5 :

FL(10.5) = 46.656(10.5) - 73.404

FL(10.5) = 416.484 thousand dollars ------> ANSWER

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