Explanation Given: The marginal cost of extracting x tons of copper ore from a mine is 0.6 + 0.008 x in thousands of dollars per ton. The extracting amount of copper varies from a = 0 and b = 50 units for the first 50 tons. The extracting amount of copper varies from a = 50 units and b = 100 units for the next 50 tons. Start-up cost is $ 100 , 000 . Theorem used: Net change theorem: The integral of a rate of change is the net change is given by, ∫ a b F ′ ( x ) d x = F ( b ) − F ( a ) (1) Here, the lower limit is a and the upper limit is b . Calculation: Apply Net change Theorem as follows: The marginal cost for extraction of copper for the first 50 tons is obtained by the use of Net change Theorem as follows. Modify Equation (1) for the calculation of cost. ∫ a b C ′ ( x ) d x = C ( b ) − C ( a ) (2) Substitute 0 for a , 50 tons for b , and [ ( 0.6 + 0.008 x ) × 1 , 000 ] for C ′ ( x ) in equation (2). ∫ 0 50 ( 0.6 + 0.008 x ) 1 , 000 d x = C ( 50 ) − C ( 0 ) C ( 50 ) = 1 , 000 ∫ 0 50 ( 0.6 + 0.008 x ) d x + C ( 0 ) (3) Substitute $ 100 , 000 for C ( 0 ) in equation (3). C ( 50 ) = 1 , 000 ( 0.6 x + 0.008 x 2 2 ) 0 50 + $ 100 , 000 = 1 , 000 × [ ( 0

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Chapter 8.4, Problem 3E

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To calculate: The cost of extracting the first 50 tons of copper and to find the cost of extracting the copper from 50 tons to 100 tons.

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Chapter 8.4, Problem 2E

Chapter 8.4, Problem 4E

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The cost of extracting the first 50 tons of copper and to find the cost of extracting the copper from 50 tons to 100 tons.

Solution Summary: The author calculates the cost of extracting the first 50 tons of copper from a mine by using the Net change theorem.

To calculate: The cost of extracting the first 50 tons of copper and to find the cost of extracting the copper from 50 tons to 100 tons.

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Chapter 8 Solutions