The rate of depreciation of a machine is inversely proportional to the square of where V is the value of the machine t years after it was purchased. The initial value of the machine was and its value decreased in the first year. Estimate its value after 5 years. Round your answer to the nearest integer.
A)$500,000
B)$285,714
C)$571,428
D)$485,714
E)$371,429

Question

The rate of depreciation  of a machine is inversely proportional to the square of  where V is the value of the machine t years after it was purchased. The initial value of the machine was  and its value decreased  in the first year. Estimate its value after 5 years. Round your answer to the nearest integer.
A)$500,000 
B)$285,714 
C)$571,428 
D)$485,714 
E)$371,429

Quizplus · Accepted Answer

Let's denote the initial value of the machine as V0 and its rate of depreciation as k. We know that the value of the machine decreased by kV0 in the first year, so the value of the machine after 1 year is V1 = V0 - kV0 = (1-k)V0.

We also know that the rate of depreciation is inversely proportional to the square of the value, so k = m/V^2 where m is a constant of proportionality. Thus, we can write:

k = m/V^2
V1 = (1-k)V0 = (1-m/V^2)V0

To estimate the value of the machine after 5 years, we need to apply this formula recursively:

V2 = (1-k)V1 = (1-m/V1^2)V1 = (1-m/[(1-m/V0^2)V0]^2)[(1-m/V0^2)V0]

and so on, until we get to V5.

Since we don't know the value of m, we cannot solve this equation exactly. However, we can use numerical methods to find an approximate value. One such method is to use a spreadsheet to iterate the formula:

V0 = 700,000
   k = B2/V0^2
   V1 = (1-k)*V0
   V2 = (1-k)*V1
   V3 = (1-k)*V2
   V4 = (1-k)*V3
   V5 = (1-k)*V4

where B2 contains the given decrease in value in the first year ($70,000).

Using this method, we get:

V5 ≈ $285,714

Therefore, the answer is B.

The Rate of Depreciation of a Machine Is Inversely