Solve the problem.
-The cost of manufacturing a particular videotape is where x is the number of tapes produced. The average cost per tape, denoted by is found by dividing C(x) by x.
A) 14
B) 6
C) 10
D) Does not exist

Question

Solve the problem.
-The cost of manufacturing a particular videotape is  where x is the number of tapes produced. The average cost per tape, denoted by  is found by dividing C(x) by x. 
A) 14 
B) 6 
C) 10 
D) Does not exist

Quizplus · Accepted Answer

The average cost per tape is found by dividing the total cost by the number of tapes produced. Therefore, the expression for the average cost per tape is:
(11ebb6d7_48c6_6f00_b437_bd9e0d4a58b1_TB8593_11)/(x)

To see whether this limit exists, we can take the limit as x approaches infinity. Using L'Hopital's Rule, we get:
lim x->∞ (11ebb6d7_48c6_6f00_b437_bd9e0d4a58b1_TB8593_11)/(x) = lim x->∞ (0)/(1) = 0

Since the limit exists (and equals 0), the average cost per tape approaches 0 as the number of tapes produced increases without bound. Therefore, the answer is choice C, 10. The average cost per tape is found by dividing C(x) by x, so we have:
11ebb6d7_48c6_6f01_b437_71b7bcf7abae_TB8593_11 = (11ebb6d7_48c6_6f00_b437_bd9e0d4a58b1_TB8593_11)/x
Therefore, as x increases, the denominator gets larger and the average cost per tape gets smaller. So, the minimum average cost per tape will occur when x is the maximum, which is not given in the problem. However, we know that the cost of manufacturing a single tape (x = 1) is 10. Therefore, the average cost per tape when only one tape is produced is C(1)/1 = 10, which is choice C.

Solve the Problem. -The Cost of Manufacturing a Particular Videotape Is Where