Question

A company's total cost, in millions of dollars, is given by C(t) = 140 - 30et where t = time in years. Find the marginal cost when t = 6. 0.35 million dollars per year O 0.16 million dollars per year 0.07 million dollars per year O 0.45 million dollars per year

Answer 1

We have to find the marginal cost of the function:

`C(t)=140-30e^(-t)`

for the value t=6. We remember that the marginal cost is defined as the derivative of the function of total cost. So, for finding the value of marginal cost, we will find its function:

`M(t)=C^(\prime)(t)=(d)/(\differentialDt t)(140-30e^(-t))`

Then, using the properties of differentiation,

`\begin{gathered} (d)/(\differentialDt t)(140-30e^(-t)) \\ =(d)/(\differentialDt t)(140)-(d)/(\differentialDt t)(30e^{-t^{}}) \\ =0-30(d)/(\differentialDt t)(e^(-t)) \\ =-30(d)/(\differentialDt t)(e^(-t)) \end{gathered}`

And then, for finding the last derivative, we use the chain rule:

`=-30(-1)e^(-t)=30e^(-t)`

This means that our marginal cost function is:

`M(t)=30e^(-t)`

Finding the value when t=6

We just have to find the value M(6), by replacing t by 6, as shown:

`M(6)=30e^{-6^{}}=(30)/(e^6)=0.0743625653\approx0.07`

This means that the marginal cost when t=6 is 0.07 million dollars per year.