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marginal condition

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A cell phone distributor can import cell phones at the world price of \$110 per cell phone. The distributor can also have these phones produced within his or her own company at two production lines. Production Line 1 has a total cost function C1= .0020 X^12, while Production Line 2 has a total cost function C2=.001X^22, where C1 (C2) are the total production costs for line 1(line 2) and X1(X2) denote the production levels for line 1 (line 2). The distributor wants to obtain 200,000 cell phones to distribute to local customers. But he or she wants to diversify the source mix among imports and the two production lines in order to minimize costs. Incurring more than the minimum cost would reduce the distributor's profit margin when he or she turns around and sells the phones. Thus, the distributor has hired you to determine the number of phones that should be imported, produced on Line 1, and produced on Line 2, to minimize the total cost of obtaining the 200,000 cell phones.

Find the following:

(a)Set this up as a constrained minimization problem, use calculus to determine the lowest-cost combination of imports and production on Lines 1 and Lines 2 to satisfy the distributor's 200,000 cell phone target. What is the marginal condition that results?

(b)What is the total cost of obtaining the cell phones based on the lowest cost combination of imports and production on lines 1 and 2 as you determined in part (a)?

(c)What is the total cost to the distributor if all 200,000 cell phones are produced on lines 1 and lines 2, where the production shares between line 1 and line 2 are distributed to minimize the total cost of 200,000 by these two sources alone? What is the cost savings of the lowest cost solution in (b) compared with totally producing the cell phones domestically on lines 1 and 2?

(d) What is the total cost for the distributor if all 200,000 cell phones are imported? What is the cost savings of the lowest cost solution in (b) compared with the cost of importing all cell phones?

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Solution Preview

Please can you help with some hints, if you can't help with final solution. Thank you!

A cell phone distributor can import cell phones at the world price of \$110 per cell phone. The distributor can also have these phones produced within his or her own company at two production lines. Production Line 1 has a total cost function C1= .0020 X^12, while Production Line 2 has a total cost function C2=.001X^22, where C1 (C2) are the total production costs for line 1(line 2) and X1(X2) denote the production levels for line 1 (line 2). The distributor wants to obtain 200,000 cell phones to distribute to local customers. But he or she wants to diversify the source mix among imports and the two production ...

Solution Summary

The marginal condition is found.

\$2.19