Independent solution

How to solve this Varying Force Accumulation Time question

Setup

Setup

For a time-varying force, the accumulation from 0 to t is the exponential of its integral. Set that factor equal to the required wealth multiple 2,000/500 = 4.

2000500=exp⁡(∫0tr2100(3+r3/150)dr)\frac{2000}{500}=\exp\left(\int_0^t\frac{r^2}{100(3+r^3/150)}dr\right)

Model

Model

The substitution u = 3 + r³/150 reduces the force integral to one-half of a logarithm, giving 4 = (1 + t³/450) to the power 1/2.

4=(1+t3450)1/24=\left(1+\frac{t^3}{450}\right)^{1/2}

Compute

Compute

Squaring and solving yields t³ = 6,750, so t = 18.8988 years, or 18.9.

t=(6750)1/3=18.8988t=(6750)^{1/3}=18.8988

Answer

Answer

The calculation gives 18.9 for varying force accumulation time, matching published choice E.

t=18.9(E)\boxed{t=18.9\quad\text{(E)}}