The rate of depreciation dv / dt of a machine is inversely proportional to the square of t + 1, where v is the value of the machine t years after it was purchased. the initial value of the machine was $500,000, and its value decreased $200,000 in the first year. estimate its value to the nearest whole number after 6 years.

The solution for this problem is:
dV/dt = r/(t + 1)², V(0) = $500000, V(1) = $500000 - $200000 = $300000 
∫dV = r∫ 1/(t + 1)² dt 
V(t) = -r/(t + 1) + C 
500000 = -r/(0 + 1) + C 
400000 = -r/(1 + 1) + C 
C = 300000, r = -100000 
V(t) = 100000/(t + 1) + 300000 
V(6) = 100000/(6+ 1) + 300000 
V(6) = 14285.7143 + 300000

V(6) = $314285.71