Directions: Answer each question, show your work in full detail, and clearly sta

Question

Directions: Answer each question, show your work in full detail, and clearly state your final answer. Explanations should be given in complete sentences. Graphs (if required) should be fully labeled. Proper notation and units (where applicable) are required. . Problem 1: (15 points) A company has determined that the demand function for a certain couch is given by p 2700-0752, where p is the price per couch, and z is the number of couches sold. The fixed costs associated with producing a line of couches is $760, 000, and each couch costs $360 to make. Find functions to represent the revenue and the total cost, then find a function for profit. Determine how many couches should be manufactured and sold in order to maximize profit.

Explanation / Answer

The demand function for a couch is given by p= 2700-0.75x, where p is the price per couch and x is the no. of couches sold. Further, the fixed costs are $ 760000 and cost of making a couch is $ 360.

The revenue function is R(x) = px = (2700-0.75x)x = 2700x -0.75x2.

The total cost function is C(x) = 760000+360x.

The profit function is P(x) = R(x)-C(x) = 2700x -0.75x2 -760000 -360x = 2340x-0.75x2 -760000.

If the profit is to be maximum, then we must have dP/dx = 0 and also d2P/dx2 should be negative. Here, dP/dx = 2340-1.50x so that if dP/dx = 0, then 2340-1.50x = 0 so that x = 2340/1.5 = 1560. Also, d2P/dx2 =-1.50 which is Always negative regardless of the value of x.

Hence, the profit is maximum if 1560 couches are manufactured and sold.

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