Chapter 6.7, Problem 11E

To determine

To Calculate: The amount of capital formation in a given duration when net investment flow rate is given.

Answer to Problem 11E

The capital formation from the fourth to the eighth year is $\$9.75 million$ .

Explanation of Solution

Given Information: Net investment flow rate is $\$\sqrt{t}\times {10}^6$ per year and the duration is from the fourth to the eighth year.

Concept Used: According to the Net Change Theorem, the integral of the rate of change is the net change, that is

  $\underset{a}{\overset{b}{{\displaystyle \int }}}F\text{'}\left(x\right)dx=F\left(b\right)-F\left(a\right)$

Calculation: In the Net Change Theorem, replace $F^{\prime }\left(x\right)$ with the ‘net investment flow’ rate $f\text{'}(x)$ and the net change with ‘capital formation’, that is $f\left(8\right)-f\left(4\right)$ , to write

  $CapitalFormatio{n}_{4\to 8}=\underset{4}{\overset{8}{{\displaystyle \int }}}\sqrt{t}dt$

  $={\frac{2}{3}{t}^{\frac{3}{2}}]}_4^8=\frac{2}{3}\left(\sqrt{512}-\sqrt{64}\right)$

  $=9.75$

Therefore, the capital formation from the fourth to the eighth year is $\$9.75 million$