Question

The resale value of a certain industrial machine decreases over a 8-year period at a rate that changes with time. When the machine is x years old, the rate at which its value is changing is 200(x - 8) dollars per year. By how much does the machine depreciate during the fifth year

Answer 1

Answer: The machine depreciates during the fifth year by $4000.

Explanation:

Given: The resale value of a certain industrial machine decreases over a 8-year period at a rate that changes with time.

When the machine is x years old, the rate at which its value is changing is 200(x - 8) dollars per year.

Then, the machine depreciates A(x) during the fifth year as

`A(x) =\int^(5)_1200(x - 8)\ dx\\\\=200|(x^2)/(2)-8x|^(5)_1\\\\=200[(5^2)/(2)-(1^2)/(2)-8(5)+8(1)]\\\\=200 [12-32]\\\\=200(-20)=-4000`

Hence, the machine depreciates during the fifth year by $4000.