This question is incomplete, the complete question is;
Your company is planning to air a number of television commercials during a television network's presentation of the an awards show. The network is charging your company $1.9 million per 30-second spot. Additional fixed costs (development and personnel costs) amount to $500,000, and the network has agreed to provide a discount of $160,000√x for x television spots.
Required:
Write down the cost function C, marginal cost function C’, and average cost function
Answer:
- The the cost function is 500,000 + 1,900,000x - 160,000√x
- the marginal cost function is 1,900,000 - (80000 /√x )
- The average cost function is 1,900,000 + [ 500,000 / x ] - [ 160,000 / √x ]
Step-by-step explanation:
Given the data in the question;
cost per spot = $1.9 million
Additional cost = $500,000
discount = $160,000√x
Let C(x) represent the cost ;
Cost x television spot = cost per spot × Number pf spots
Cost x television spot = $1.9 million × x
Cost x television spot = $1,900,000x
Now, the television set total cost will be;
C(x) = television cost + additional cost - discount
C(x) = 500,000 + 1,900,000x - 160,000√x
Therefore, The the cost function is 500,000 + 1,900,000x - 160,000√x
Marginal Cost Function;
Cost function C(x) = 500,000 + 1,900,000x - 160,000√x
we differentiate with respect to x
C'(x) = d/dx( 500,000 + 1,900,000x - 160,000√x )
= d/dx( 500000 ) + 1,900,000d/dx -160,000 d/d( √x )
= 0+ 1,900,000(1) -160,000( 1 / 2√x )
= 1,900,000 - (160,000 / 2√x )
= 1,900,000 - (80000 /√x )
Therefore, the marginal cost function is 1,900,000 - (80000 /√x )
Average cost function;
Average cost function = C(x) / x
we substitute
Average cost function = [500,000 + 1,900,000x - 160,000√x] / x
= [500,000 / x ] + [1,900,000x / x ] - [ 160,000√x / x ]
= [ 500,000 / x ] + 1,900,000 - [ 160,000√x / x ]
= 1,900,000 + [ 500,000 / x ] - [ 160,000 / √x ]
Therefore, The average cost function is 1,900,000 + [ 500,000 / x ] - [ 160,000 / √x ]
