A town obtains its water supply from two wells, W1 and W2. The cost of securing

Question

A town obtains its water supply from two wells, W1 and W2. The cost of securing water from each well is defined as follows:
CostW1 = 13 + 4Q1 + (Q1)2
CostW2 = 26 + 8Q2 + 4(Q2)2
Where Q1 and Q2 are the amount of water in million gallons per day (mgd) from each well.
The town needs a total of 2 mgd. The capacity of well 1 is 5 million gallons per day, and that of well 2 is 10 mgd. Using calculus (hint: calculus with substitution), determine the optimal amount of water that will be drawn from each well if the town wants to minimize total costs.

Explanation / Answer

Total cost C = CostW1+CostW2 = (13 + 4Q1 + Q12) + (26 + 8Q2 + 4Q22) = 39 + 4Q1 + 8Q2 + Q12 + 4Q22

Q1 + Q2 = 2.

Q2 = 2 - Q1

So, C = 39 + 4Q1 + 8*(2-Q1) + Q12 + 4*(2-Q1)2

C = 63 - 20Q1 + 5Q12

For minimum cost, dC/dQ1 = 0

-20 + 10Q1 = 0

Q1 = 2 mgd,

Hence, Q2 = 2-2 = 0

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