The cost of producing x units of a product is C = 60x + 1350. For one week management determined the number of units produced at the end of t hours during an eight-hour shift. The average values of x for the week are shown in the table.

t 0 1 2 3 4 5 6 7 8
x 0 16 60 130 205 271 336 384 392
(a) Use a graphing utility to fit a cubic model to the data. (Round your coefficients to three decimal places.)
x(t) = -1.637t^3+ 19.312t^2 -0.508t-0.616


(b) Use the Chain Rule to find dC/dt. (Round your coefficients to three decimal places.)

How do I complete part b?

dC/dt = dC/dx dx/dt = 60 dx/dt

= 60 [ 3(-1.637) t^2 etc

To complete part b, we need to find the derivative of the cost function, C, with respect to time, t. This is represented by dC/dt.

The given cost function is C = 60x + 1350, where x represents the number of units produced. We can rewrite this equation in terms of t using the cubic model for x, which is x(t) = -1.637t^3 + 19.312t^2 - 0.508t - 0.616.

Substituting the value of x(t) into the cost function, we get:

C(t) = 60(-1.637t^3 + 19.312t^2 - 0.508t - 0.616) + 1350.

Expanding and simplifying the equation, we have:

C(t) = -98.22t^3 + 1158.72t^2 - 30.48t - 36.96 + 1350.

C(t) = -98.22t^3 + 1158.72t^2 - 30.48t + 1313.04.

Now we can find the derivative of C(t) with respect to t, i.e., dC/dt. We can use the power rule for derivatives, where for any term of the form a*t^n, the derivative is given by n*a*t^(n-1).

Taking the derivative of C(t) term by term, we get:

dC/dt = -294.66t^2 + 2317.44t - 30.48.

Therefore, dC/dt = -294.66t^2 + 2317.44t - 30.48 is the derivative of the cost function with respect to time.