The management of Acrosonic plans to market the ElectroStat, an electrostatic speaker system. The marketing department has determined that the demand for these speakers is , where p denotes the speaker's unit price (in dollars) and x denotes the quantity demanded.
Acrosonic's production department estimates that the total cost (in dollars) incurred in manufacturing x ElectroStat speaker systems in the first year of production will be .

Find the profit function P and select the graph of the profit function.

Question

The management of Acrosonic plans to market the ElectroStat, an electrostatic speaker system. The marketing department has determined that the demand for these speakers is  , where p denotes the speaker's unit price (in dollars) and x denotes the quantity demanded. ​
Acrosonic's production department estimates that the total cost (in dollars) incurred in manufacturing x ElectroStat speaker systems in the first year of production will be  .
​
Find the profit function P and select the graph of the profit function.
​

Quizplus · Accepted Answer

The profit function is given by: 
P(x) = revenue - cost 
         = px - C(x)

From the given information, we know that the demand function is: 
x = 11eb651f_cb6e_5900_b0a1_db19a3713e0a_TB8255_11 / p

And the total cost function is: 
C(x) = 11eb651f_cb6e_5901_b0a1_57b7c90a1138_TB8255_11

Substituting the total cost function and the demand function into the profit function, we get: 
P(x) = px - 11eb651f_cb6e_5901_b0a1_57b7c90a1138_TB8255_11 / x

Simplifying this expression by common denominator: 
P(x) = (px^2 - 11eb651f_cb6e_5901_b0a1_57b7c90a1138_TB8255_11) / x

This is a quadratic function in x, with a negative coefficient on the squared term. Therefore, the graph of the profit function will be a parabola opening downwards.

The correct choice is B, as it is the only option with a downward parabolic graph.

The Management of Acrosonic Plans to Market the ElectroStat, an Electrostatic