A company buys a machine for $20 million. Four years later, the machine is worth $12 million.

Question

A company buys a machine for $20 million. Four years later, the machine is worth $12 million. Assume the decline in value follows the exponential model y=y0e^kt. Find the exponential model. Use the model to determine the machine's value in another two years.

Answer 1

y = value of machine (in millions) t years after purchase

 

y = y₀e^kt

 

When t = 0, y₀e^k(0) = 20, so y₀ = 20

 

y = 20e^kt

 

When t = 4, y = 12.   So, 12 = 20e^4k

 

                                            0.6 = e^4k

 

                                          ln(0.6) = ln(e^4k)

 

                                          ln(0.6) = 4k         k = -0.127706405

 

y = 20e^{-0.127706405t}

 

So, when t = 6, y = 20e^{-0.127706405(6)} = 9.29516 million $