Solve the problem.
-The demand equation for a certain item is and the cost equation is C(x) = 7,000 + 4x. Find the marginal profit at a production level of 3,000 and interpret the result.
A)$14; at the 3,000 level of production,profit will increase by approximately $14 for each unit increase in production.
B)$16; at the 3,000 level of production,profit will increase by approximately $16 for each unit increase in production.
C)$7; at the 3,000 level of production,profit will increase by approximately $7 for each unit increase in production.
D)$4; at the 3,000 level of production,profit will increase by approximately $4 for each unit increase in production.

Question

Solve the problem.
-The demand equation for a certain item is  and the cost equation is C(x) = 7,000 + 4x. Find the marginal profit at a production level of 3,000 and interpret the result.
A)$14; at the 3,000 level of production,profit will increase by approximately $14 for each unit increase in production. 
B)$16; at the 3,000 level of production,profit will increase by approximately $16 for each unit increase in production. 
C)$7; at the 3,000 level of production,profit will increase by approximately $7 for each unit increase in production. 
D)$4; at the 3,000 level of production,profit will increase by approximately $4 for each unit increase in production.

Quizplus · Accepted Answer

The revenue equation is given as R(x) = 11ec8a78_07af_c0eb_bfc5_35fcc3e64cab_TB6722_11. 
The profit function is given by P(x) = R(x) - C(x). 
At a production level of 3,000, the revenue is R(3,000) = 11ec8a78_07af_c0eb_bfc5_35fcc3e64cab_TB6722_41 and the cost is C(3,000) = 7,000 + 4(3,000) = 19,000. Therefore, the profit at a production level of 3,000 is:
P(3,000) = R(3,000) - C(3,000) = (11ec8a78_07af_c0eb_bfc5_35fcc3e64cab_TB6722_41) - 19,000.
To find the marginal profit, we need to find the derivative of the profit function:
P'(x) = R'(x) - C'(x)
Since the demand equation is given, we can find the revenue function: 
R(x) = 11ec8a78_07af_c0eb_bfc5_35fcc3e64cab_TB6722_11 = (45 - 0.001x)x = 45x - 0.001x^2
Then, we have:
R'(x) = 45 - 0.002x
C'(x) = 4
So, P'(x) = 45 - 0.002x - 4 = 41 - 0.002x
At a production level of 3,000, the marginal profit is:
P'(3,000) = 41 - 0.002(3,000) = 35
Therefore, for each unit increase in production, profit will increase by approximately $4 ($35 divided by 9,000 units).

Solve the Problem. -The Demand Equation for a Certain Item Is and Is