Calculus integration question

Question

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 $800,000, and its value decreased $200,000 in the first year. Estimate its value to the nearest whole number after 4 years.

Answer 1

Hi Abigail,

 

dV/dt  = - k/(t +1)²   where k is the constant of proportionality.

V(t=0) = $800,000,

Since dV/dt|_t=1 =$200.00

200000 = -k/(1+1)² = -k/4  we get  k= -800,000

dV = [800,000/(t+1)²] dt

V(t) = 800000 ∫(t + 1)^-2 dt = -800000/(t+1) + C

V(t) = 800000/(t+1) + C

V(0) = 800000/(0+1) + C

800000 = 800000 + C  so C =0

V(t)=800,000/(t+1)

V(4) =800000/(4+1) =$160,000