Question Problem 3 (2 points): If the rate of net investment flow is given by I(t) = 2002, calculate: a/The capital formation from the end of the second year to the end of the sixth year; b/The number of years required before the capital stock exceeds $200 000

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Transcribed Image Text: Problem 3 (2 points): If the rate of net investment flow is given by I(t) = 2002, calculate: a/The capital formation from the end of the second year to the end of the sixth year; b/The number of years required before the capital stock exceeds $200 000

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Sof.". Given Rate of net investastruent flow.I(t)=200 e^{0.2 t}(a) total capital formation from 2^{\text {nd }} year to 6^{\text {thy }} yearSo t=3 to 6\begin{aligned}\text { Capital formation } & =\int_{3}^{6} I(t) d t \\& =\int_{3}^{6} 200 e^{0.2 t} d t \\& =\left[\frac{200}{0.2} \times e^{0.2 t}\right]_{3}^{6}=\frac{200}{0.2}\left[e^{0.2 \times 6}-e^{0.2 \times 3}\right] \\\text { total Capital } & =1000\left[e^{1.2}-e^{0.6}\right]=1000 \times 1.4979 \\\text { formation } & =1497.9 \$\end{aligned}(b) Here we supposed x year requires to Capital Exceeds \$ 200,000.\Rightarrow 200,000=\int_{0}^{n} I(t) d tput value of I(t) as given\begin{array}{l}\Rightarrow 200,000=\int_{0}^{n} 200 e^{0.2 t} d t \\\Rightarrow 200,000=\frac{200}{0.2}\left[e^{0.2 t}\right]_{0}^{n}\end{array}\begin{aligned}\Rightarrow \quad 200,000 & =1000\left[e^{0.2 n}-e^{0}\right] \\\Rightarrow \quad 200 & =e^{0.2 n}-1 \quad\left\{e^{0}=1\right\} \\\Rightarrow \quad e^{0.2 n} & =201 \\0.2 n & =\ln 201 \quad\{\ln 201=5.303\} \\S 0 n & =\frac{5.303}{0.2}=26.51 \text { Years }\end{aligned}So for the capital stocks exceeds \$ 200,000 total 26.51 Years required. Is there any doubt let me know, thanks ud83dudc4d ...