Rate of price depreciation

Question

The rate of depreciation $\frac{dV}{dt}$ 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 is $500,000$ and its value decreased $ 100,000$ in the first year. Estimate the value after four years.

The function is $$\frac{dV}{dt}=\frac{K}{(t+1)^2} $$

We can integrate with respect to $t$, which is time, and get $$V(t)=-\frac{K}{t+1}+C$$ We also know the values at $V(0)$ and $V(1)$ which are $$V(0)=500,000$$ $$V(1)=400,000$$ Now setting up the equation for each value gives us $$V(0)=500,000= -\frac{K}{1}+C$$ $$V(1)=400,000=-\frac{K}{2}+C $$ I have solved this equation before and gotten $K=-200,000$ and $C=300,000$ but I cannot remember how I arrived at these answers.

Answer 1

Take the top equation and subtract the bottom one for:

$$100,000=-\frac{K}{2} \implies K=-200,000.$$

Then just sub back into the top equation:

$$500,000=-\frac{-200,000}{1}+C$$ $$500,000=200,000+C \implies C=300,000.$$

Answer 2

Multiplying the second equation by $2$, you have the pair of equations $$ \left\{ \begin{array}{ccc} -K & + C &= 500~000 \\ -K & +2C &= 800~000 \end{array} \right. $$ Substracting the first from the second, we get $C=300~000$. Then we only have to calculate (using the first equation) $$ K = C - 500~000 = -200~000 $$ It's that easy.

Answer 3

rearranging we can get: $$K=C-500,000$$ $$K=2(C-400,000)$$ now making them equal we get: $$-800,000+C=-500,000$$ and so: $$C=300,000$$ now we want to substitute this back into one of the original equations: $$K=(300,000)-500,000=-200,000$$