# The rate of depreciation dVdt 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. - Mathematics and Statistics

Sum

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 ₹ 8,00,000 and its value decreased ₹1,00,000 in the first year. Find its value after 6 years.

#### Solution

According to the given condition,

(dV)/dt ∝ 1/(t+1)^2

∴ (dV)/dt = (-k)/(t+1)^2 …[Negative sign indicates disintegration]

∴ dV =  (-kdt)/(t+1)^2

Integrating on both sides, we get

int dV = - k int dt/(t+1)^2

∴ V = k/(t+1) + c

when t = 0, V = 8,00,000

∴ 8,00,000 = k/((0+1)) +c

∴ 8,00,000 = k + c …(i)

when t = 1, V = 7,00,000

∴ 7,00,000 = k /((1 +1)) + c

∴ 7,00,000 = k/ 2 + c …(ii)

From (i) – (ii), we get

1,00,000 = k /2

∴ k = 2,00,000  …(iii)

Substituting (iii) in (i), we get

c = 6,00,000  …(iv)

when t = 6, we get

V = k/((6+ 1)) + c

=(2,00,000 )/7 + 6,00,000

= 6,28,571.4286

≈6,28,571

∴ Value of the machine after 6 years is ₹ 6,28,571.

Concept: Application of Differential Equations
Is there an error in this question or solution?

#### APPEARS IN

Balbharati Mathematics and Statistics 1 (Commerce) 12th Standard HSC Maharashtra State Board
Chapter 8 Differential Equation and Applications
Exercise 8.6 | Q 5 | Page 170