Question

The total cost of producing a decorative item is C = 22.87x2 - 0.08x3 where x = number of units produced. Find the value of x to achieve a minimum average unit cost. Round your answer to two decimal places. 

          The total cost of producing a decorative item is C = 22.87x2 - 0.08x3 where x = number of units produced. Find the value of x to achieve a minimum average unit cost. Round your answer to two decimal places.

Added by Christopher A.

Calculus: Early Transcendentals
Calculus: Early Transcendentals
James Stewart 8th Edition
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The total cost of producing a decorative item is C = 22.87x2 - 0.08x3 where x = number of units produced. Find the value of x to achieve a minimum average unit cost. Round your answer to two decimal places.
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Transcript

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00:01 Here we're given that the average cost for a product is given by c of x is equal to 2x plus 54 plus 98 over x and this can be written as 2x plus 54 plus 98 x to the power of minus 1 according to the laws of indices and we want to find how many units must be produced in order is to minimize the average cost.
00:31 Now this is given us the average cost and we want to minimize it.
00:39 Now we know that at minimum or maximum we have c prime of x being equal to zero.
00:48 So first we have to find the marginal average cost, which is c prime of x with the first derivative of the cost function.
00:56 And here we have the derivative of 2x.
01:01 2 plus the derivative of 54 since there is no x term there it's a constant get a 0 and here the derivative of 98 x to the power of minus 1 we get minus 1 by 98 x to the of minus 2 and this can be written as 2 minus 98 over x squared now equating this to 0 because at critical points we have have our derivative equal to zero and solving for x we have two is equal to 98 over x squared cross multiplying we get 2x squared is equal to 98 to are in both sides by 2 we get x squared is equal to 49 and taking the root we get x is equal to plus or minus 7 however negative 7 does not make sense for the number of units so we just take the principal value which is 7...
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