Suppose a company has flxed costs of $51,000 and variable cost per uni of 2x +44

Question

Suppose a company has flxed costs of $51,000 and variable cost per uni of 2x +444 dollars, where x Is the total number of units produced. Suppose further that the selling price of its product Is 1978 x dollars per unit. (a) Find the break-even points. (Enter your answers as a comma-separated list.) (b) Find the maximum revenue. (Round your answer to the nearest cent.) (c) Form the profit function (x) from the cost and revenue functions P(x) - Find maximum profit. (d) What price will maximize the profit? (Round your answer to the nearest cent.)

Explanation / Answer

x = total number of units

C(x) = total cost = 1000 + (2/5x+444)x

R(x) = total revenue = (1978 - 3/5x) x

(a)

break even points are where cost and revenue are same

C(x) = R(x)

1000 + (2/5x+444)x = (1978 - 3/5x) x

solve using by creating it a quadratic equation

x^2-1534x + 1000 = 0

x = 1533.39

so it means approximately on producing 1533 units, breakeven will happen.

(note the other root is negative)

(b)

R(x) = total revenue = (1978 - 3/5x) x

revenue will be maximum at point where first derivative is 0.

R'(x) = 1978 - 6/5x = 0

x = 1978*5/6 = 1648.33 ~ 1648 units

(c)

P(x) = R(x) - C(x)

P(x) = -(x^2-1534x + 1000)

(d)

profit will be maximum at point where first derivative of profit function is 0 .

P(x) = -(x^2-1534x + 1000)

P'(x) = 2x - 1534 = 0

x = 1534/2 = 767

so on producing 767 units, profit will be maximum  

I am answering multiple questions here, while the Chegg rules states that I am obliged to answer only 1 qs per post. Please take this kindness into consideration and upvote the answer.

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