Your company is planning to air a number of television commercials during the ABC television network's presentation of the Academy Awards. ABC is charging your company $725,000 per 30-second spot. Additional fixed costs (development and personnel costs) amount to $400,000, and the network has agreed to provide a discount of $D ( x ) = 25,000 \sqrt { x }$ for x television spots. Compute marginal cost $C ^ { \prime } ( 5 )$ and average cost $\bar { C } ( 5 )$ .
A) $C ^ { \prime } ( 5 ) = \$ 718,410$ per spot; $\bar { C } ( 5 ) = \$ 793,820$ per spot
B) $C ^ { \prime } ( 5 ) = \$ 793,820$ per spot; $\bar { C } ( 5 ) = \$ 756,615$ per spot
C) $C ^ { \prime } ( 5 ) = \$ 719,410$ per spot; $\bar { C } ( 5 ) = \$ 793,720$ per spot
D) $C ^ { \prime } ( 5 ) = \$ 793,820$ per spot; $\bar { C } ( 5 ) = \$ 719,410$ per spot
E) $C ^ { \prime } ( 5 ) = \$ 719,410$ per spot; $\bar { C } ( 5 ) = \$ 793,820$ per spot

Question

Your company is planning to air a number of television commercials during the ABC television network's presentation of the Academy Awards. ABC is charging your company $725,000 per 30-second spot. Additional fixed costs (development and personnel costs) amount to $400,000, and the network has agreed to provide a discount of $D ( x ) = 25,000 \sqrt { x }$ for x television spots. Compute marginal cost $C ^ { \prime } ( 5 )$ and average cost $\bar { C } ( 5 )$ .  
A) $C ^ { \prime } ( 5 ) = \$ 718,410$ per spot; $\bar { C } ( 5 ) = \$ 793,820$ per spot
B) $C ^ { \prime } ( 5 ) = \$ 793,820$ per spot; $\bar { C } ( 5 ) = \$ 756,615$ per spot
C) $C ^ { \prime } ( 5 ) = \$ 719,410$ per spot; $\bar { C } ( 5 ) = \$ 793,720$ per spot
D) $C ^ { \prime } ( 5 ) = \$ 793,820$ per spot; $\bar { C } ( 5 ) = \$ 719,410$ per spot
E) $C ^ { \prime } ( 5 ) = \$ 719,410$ per spot; $\bar { C } ( 5 ) = \$ 793,820$ per spot

Quizplus · Accepted Answer

The total cost function $C(x)$ for airing $x$ television spots includes the cost per spot, the fixed costs, and the discount provided. The cost per spot is $725,000, the fixed costs are $400,000, and the discount function is $D(x) = 25,000\sqrt{x}$. Therefore, the total cost function is $C(x) = 725,000x + 400,000 - 25,000\sqrt{x}$. The marginal cost $C'(x)$ is the derivative of the total cost function with respect to $x$, which gives us the cost of airing an additional spot after $x$ spots have already been aired. The average cost $\bar{C}(x)$ is the total cost divided by the number of spots, $x$.To find $C'(x)$, we differentiate $C(x)$ with respect to $x$:$$C'(x) = 725,000 - \frac{25,000}{2\sqrt{x}}$$Plugging in $x = 5$:$$C'(5) = 725,000 - \frac{25,000}{2\sqrt{5}} = 725,000 - \frac{25,000}{2 \times 2.236} \approx 725,000 - 5,590 = 719,410$$To find $\bar{C}(5)$, we calculate $C(5)$ and then divide by 5:$$C(5) = 725,000 \times 5 + 400,000 - 25,000\sqrt{5} = 3,625,000 + 400,000 - 25,000 \times 2.236 \approx 4,025,000 - 55,900 = 3,969,100$$$$\bar{C}(5) = \frac{3,969,100}{5} = 793,820$$Therefore, the correct answer is $C'(5) = \$719,410$ per spot; $\bar{C}(5) = \$793,820$ per spot.

Your Company Is Planning to Air a Number of Television D(x)=25,000xD ( x ) = 25,000 \sqrt { x }D(x)=25,000x​

Your Company Is Planning to Air a Number of Television $D ( x ) = 25,000 \sqrt { x }$