022 Sample Final A, Problem 7

Find the present value of the income stream $Y=20+30x$ from now until 5 years from now, given an interest rate $r=10\%.$
(Note: Once you plug in the limits of integration, you are finished; you do not need to simplify our answer beyond that step.)

Foundations:
The idea of an income stream is bit more complicated to set up than basic interest problems. We have two forces adjusting the balance of the account: the income stream, usually represented as $I(t)$, which represents the desired income to be withdrawn from the account, and the interest rate paid to the account. In order to evaluate this result, we use the formula
present value = $\int _{0}^{T}I(t)e^{-rt}\,dt,$
where $T$ is the time when our stream will run out, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r} is the rate (compounded continuously) paid by the bank and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle I(t)} is the desired continuous income stream.

Solution:

There isn't much to do here, except identify that Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle T\,=\,5} , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle I(t)\,=\,20+30x} and the rate should be written as Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r\,=\,10%\,=\,0.10} . Click to the final answer to see them in the formula!

Final Answer:
Present value = Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \int_0 ^T I(t)e^{-rt}\,dt,\,=\,\int_0 ^5 (20+30x)e^{-0.10t}\,dt.}

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