Depreciation Problem

Question

In 2020, a manufacturing company purchased a new fabricating machine for $317,000. For tax purposes, the machine depreciates at a constant rate for 20 years, after which its book value is $0.

a. Define a function V(t) that expresses the book value of the machine t years after purchase. (Your answer should include V(t))

b. The company plans to replace the machine when it has depreciated to 20% of its original value. Use the function you found in part (a) to determine in what year this will occur.

Answer 1

a. Note that the depreciation occurs at a constant rate, so V(t) will be a linear equation of the form

V(t) = mt + b. m = (0-31700)/(20 - 0) = -15850. So V(t) = -15850t + b. But since we know the point (0,317000), then b = 317000. So V(t) = -15850t+317000.

b. To find when the machine will be 20% of its original value, find 0.2*317000 = 63400. Now let 63400 = V(t) to get 63400 = -15850t+317000. Solve for t => (63400-317000)/(-15850) = t = 16. 16 years after 2020 will be 2036, and that is when they will need to replace the machine.

Hope that helps!