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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

A companys total cost in millions of dollars is given by Ct 140 30et where t time in years Find the marginal cost when t 6 035 million dollars per year O 016 mi class=

Respuesta :

We have to find the marginal cost of the function:

[tex]C(t)=140-30e^{-t}[/tex]

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:

[tex]M(t)=C^{\prime}(t)=\frac{d}{\differentialDt t}(140-30e^{-t})[/tex]

Then, using the properties of differentiation,

[tex]\begin{gathered} \frac{d}{\differentialDt t}(140-30e^{-t}) \\ =\frac{d}{\differentialDt t}(140)-\frac{d}{\differentialDt t}(30e^{-t^{}}) \\ =0-30\frac{d}{\differentialDt t}(e^{-t}) \\ =-30\frac{d}{\differentialDt t}(e^{-t}) \end{gathered}[/tex]

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

[tex]=-30(-1)e^{-t}=30e^{-t}[/tex]

This means that our marginal cost function is:

[tex]M(t)=30e^{-t}[/tex]

Finding the value when t=6

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

[tex]M(6)=30e^{-6^{}}=\frac{30}{e^6}=0.0743625653\approx0.07[/tex]

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

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