A manufacturer estimates that its product can be produced at a total cost of C(x) = 50,000 + 100x + x3 dollars. If the manufacturer's total revenue from the sale of x units is R(x) = 3400x dollars, determine the level of production x that will maximize the profit. (Round your answer to the nearest whole number.)

The level of production x that will maximize the profit is: 22,966

Explanation:

C(x) = 50,000 + 100x + x³

R(x) = 3400x

P(x) = R(x) - C(x)

      = 3400x - [50,000 + 100x + x³]

      = 3400x - 50,000 - 100x - x³

      = 3300x - 50,000 - x³   .................... (A)

P'(x) = 3300(1) - 0 - 3x²

       = 3300 - 3x²

At a critical point, P'(x) = 0

∴   0 = 3300 - 3x²

  3x² = 3300

    x² = 1100

     x = ±

P"(x) = -6x

P() = -6 ()   < 0

by second derivative, 'P' max at    x = = 33.17 (rounds)

since x =   ,

recall that P(x) = 3300x - 50,000 - x³ from equation (A)

Therefore, Maximum Profit

P() = 3300 - 50000 -

              = 3300(33.17) - 50,000 - 33.17³

              = 109461 -50,000 - 36495.26

              = 22,965.74

Maximum profit is 22,966 to the nearest whole number

so they can see what that city is producing and to also see if they can some how get that product. like gold in africa.

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