The resale value of a machine decreases over a 10 year period at a rate that depends on the age of the machine. When the machine is x years old, the rate at which its value is changing is ₹ 2200 (x − 10) per year. Express the value of the machine as a function of its age and initial value. If the machine was originally worth ₹1,20,000, how much will it be worth when it is 10 years old?

#### Solution

Let ‘y’ be the value of the machine when machine is ‘x’ years old.

∴ According to the given condition,

`dy/dx = 2200 (x - 10)`

∴ dy = 2200(x – 10) dx

Integrating on both sides, we get

∫1 dy = 2200 ∫ (x-10) dx

∴ `y = 2200 (x^2/2 - 10x) + c`

∴ y = 1100 x^{2} – 22,000 x + c

when x = 0, y = 1,20,000

∴ 1,20,000 = 1100(0)^{2 }– 22,00(0) + c

∴ c = 1,20,000

∴ The value of the machine can be expressed as a function of it’s age as

y = 1,100x^{2} – 22,000x + 1,20,000

Initial value: when x = 0, y = 1,20,000

∴ when x = 10,

y = 1100(10)^{2} – 22,000(10) + 1,20,000

= 10,000

∴ The machine will worth ₹ 10,000 when it is 10 years old.