1. A factory for a company produces armils made out of orichalcum. The factory\'

Question

1. A factory for a company produces armils made out of orichalcum. The factory's costs per week are 5-o(t) p(t), where o(t) is the cost of orichalcum ore by the pound at time t (where t is measured in weeks from the beginning of the year) and p(t) is the number of armils produced. In the fourth week of the year the cost of orichalcum ore is 2000 dollars per pound, the factory is producing 60 armils per week. Additionally, the cost of orichalcum ore is rising by 50 dollars per week, and the factory is decreasing its production of armils by 2 a week. (That is, o(4) = 2000. p(4)-60, d(4-50. p'(4)--2) With this situation, find the exact rate at which the factory's costs per week are increasing or decreasing.

Explanation / Answer

C = 5 * o * p

Deriving :
C' = 5*o*p' + 5*o'*p
by product rule

Now, plug in the values for t = 4 :
C'(4) = 5*2000*(-2) + 5*50*60

C'(4) = -20000 + 15000

C'(4) = -5000

So, the costs per week are decreasing at
5000 dollars per week ----> ANS

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