Independent solution

How to solve this Varying Force of Interest question

Setup

Setup

Let a(t) be the accumulation from time 0 under the time-varying force. Integrating the force gives a(t) = exp(t³/300).

a(t)=exp⁡(∫0ts2/100 ds)=et3/300a(t)=\exp\left(\int_0^t s^2/100\,ds\right)=e^{t^3/300}

Model

Model

Value the balance immediately after the time-3 deposit and carry it to time 6. The equation sets the interest earned over that interval equal to the added amount X.

(100a(3)+X)(a(6)a(3)−1)=X(100a(3)+X)\left(\frac{a(6)}{a(3)}-1\right)=X

Compute

Compute

Substitution of the two accumulation factors and solution of the resulting linear equation give X = 784.61, which rounds to 785.

X=784.61X=784.61

Answer

Answer

The calculation gives 785 for varying force of interest, matching published choice E.

X=785(E)\boxed{X=785\quad\text{(E)}}