Find a formula when given cost and salvage value.

Question

A manufacturer buys a new machine costing $120000 dollars. It is estimated that the machine has a useful lifetime of ten years and a salvage value of 4000 dollars at that time. Assume the value of the machine varies linearly during this time. Find a formula for the value of the machine after t years, where 0<=t<=10. Find the value of the machine after 8 years.

Answer 1

Let f(t) be the function for the formula of machine value in t years.

We know that today it costs 120,000. Mathematically this means, at t = 0, f(t) = 120,000, or f(0) = 120,000.

Next, we know that in 10 years, it will cost 4,000 only. Again, mathematically this means, at t = 10, f(t) = 4,000, or f(10) = 4,000. 

Now any function of a line can be written as f(t) = m t + b, where one needs to determine constants m and b. From equations above we already know that

f(0) = 120, 000
m*0 + b = 120, 000
b = 120, 000

Now we know b. From the second equation we can write:

f(10) = 4,000
m*10 + 120,000 = 4,000
10 m = -116,000
m = -11,600

Finally: 
f(t) = -11,600 t + 120,000

Now using this nice formula, we can replact t with 8, i.e. t = 8, and solve for f(8):

f(8) = -11,600*8 + 120,000 = 27,200