# Thread: Time rate of change of cost problem

1. ## Time rate of change of cost problem

The cost of manufacturing x cases of umbrellas is C dollars (in thousands), where

Monthly production at t weeks is expected to be

I know that = 18(5000+80t)+4sqrt(5000+80t)+9

and that marginal cost or = 28+(2/x^.5)

I'm not sure how to find in order to solve the equation for t=2 in order to find how fast costs are rising per week.

2. Originally Posted by terrytriangle
The cost of manufacturing x cases of umbrellas is C dollars (in thousands), where

Monthly production at t weeks is expected to be

I know that = 18(5000+80t)+4sqrt(5000+80t)+9

and that marginal cost or = 28+(2/x^.5)

I'm not sure how to find in order to solve the equation for t=2 in order to find how fast costs are rising per week.
Hiya Terry by the step where you list = 18(5000+80t)+4sqrt(5000+80t)+9 , you are on the right track. Since you now have C(t), you can differentiate to find $\displaystyle \frac{dC}{dt}$ without having to worry about the chain rule with other variables.

The derivative of = 18(5000+80t)+4sqrt(5000+80t)+9 would be $\displaystyle \frac{dC}{dt}$ = 160*(5000+80t)^(-1/2) + 1440t. C'(2)=160*(5000+80(2))^(-1/2) + 1440(2)

To take the derivative of the square root of 5000+80t a chain rule must be used so
$\displaystyle \frac{d}{dt}$ (5000+80t)^(1/2)=

$\displaystyle \frac{1}{2}$ (5000+80t)^(-1/2) * 80 since 80 is the derivate of the inside term.