What rate is the cost to produce the cats changing?

Question

The cost of manufacturing stuffed yellow cats is C(x)= 1/3x² + 4x + 53 where x is the number of cats produced in thousands. The production level for the number of stuffed cats produced after t hours is given by x(t)= .2t² + .03t. At 4 hours, at what rate is the cost to produce the cats changing?

Answer 1

Hi Meagan, they are asking the rate of change of cost with time i.e. dC/dt

 

Note: C is function of x, C(x)

         x is function of t, x(t)

 

we can solve for dC/dt:

 

dC/dt = dC/dx   * dt/dx

 

dC/dx = 2/3x + 4

dx/dt = 0.4t + 0.3 

 

∴dC/dt = (2/3x +4) / (0.4t +0.3) 

 

notice there is an x term in the numerator. Replace x with x(t) = 0.2t² + 0.3t

Now:

 

dC/dt = (2/3*[0.2t2 + 0.3t]+4) / (0.4t +0.3)

 

The last step is to find dC/dt at 4 hours. We let t = 4:

 

dC/dt = 208/57 hours ≅ 3.5 hours.

 

I hope this helps you understand the process Meagan.