Business calculus optimization problem?

[brich]

I would like some help with this problem if possible:

Suppose a firm can produce both rugs and carpet, according to the following total (or per unit) cost function:
TC = 10r^2 + 11c^2 - 1rc, where:

r is rugs produced per day and c is feet of carpet produced per day. Suppose this firm must produce 10 rugs per day. To minimize per unit costs, how many feet of carpet should they produce per day?

As I understand it from the focus on derivitives in class, I would need to take the derivitave of this equation for the marginal cost and set it to zero and solve for C to get the feet of carpet per day.

When I take the derivitive of dtc/dc I get 22c-r

then 22c - 10 = 0 (sub 10 rugs per day for r and set equal to zero

c = 10/22 or .45 feet of carpet per day

That cant be correct - I must have gone wrong somewhere?
Any help would be appreciated.


Brad

 

[stapel]

You are given that r must be 10, so plug that in:

. . . . .TC = 10[10]² + 11c² - 1[10]c

. . . . .TC(c) = 1000 + 11c² - 10c

Differentiate, and solve for the value of c which minimizes TC.

If you are not sure of your answer, try graphing y = 1000 - 10x + 11x², and verifying the location of the vertex of the parabola.

Eliz.

 

[brich]

Thanks for your help

So I guess I was on the right track I just neglected to plug my value for c back into the TC(c) equation. I pulled out my 10 year old college graphing calculator (yes its been that long) and graphed it per your advice to verify my answer.

Im in grad school now trying to get the rust off so to speak. My math skills have always been somewhat poor and havent gotten better with time.........thanks for the help - much appreciated.

Brad