Question

Given the investment flow

I(t) = 2400√t

(a) Calculate the total capital formation during the first four years.

(b) Find an expression for the annual capital formation during the Nth year and hence find the first year in which the annual capital formation exceeds $4000.

Answer 1

a.

During the first four years

Total capital formation

$\int_1^{4}2400\sqrt{t}dt\\2400\frac{2t^{\frac{3}{2}}}{3}\int_1^{4}\\1600(4^{\frac{3}{2}}-1)\\1600×7\\\$11200$

b.

Annual capital formation =12×$\int_1^{n}2400\sqrt{t}dt$

Since annual capital formation exceed $400

$4000=2400\sqrt{t}\\\frac{20}{12}=\sqrt{t}\\\frac{5}{3}=\sqrt{t}\\t=\frac{25}{9}\\t=2.78 \space years$

After 2.78 years annual capital formation will exceed $4000.