The demand for a product is p = 70-0.7x where x is the quantity for the item sol

Question

The demand for a product is p = 70-0.7x where x is the quantity for the item sold at $p dollars per piece. Determine the revenue function and the domain of the function. Include a full explanation of how you found the revenue function. What price should the company charge to obtain the largest revenue? Explain how you determined this price. You must use calculus techniques in solving for this price. What is the maximum revenue? Find the marginal revenue function. Find the marginal revenue function evaluated at x= 20. Explain what this value means. Taking into account storage and shipping, it costs p = C(x) = 100 +11.25x dollars to sell x of the product in the season. Find the average cost function and the marginal average cost function. Explain the difference between the two. Find the cost, average cost and marginal average cost for 20 items. Explain the difference in these values. Find the profit function. Explain how you determined this function. What price should the company charge to get the largest seasonal profit? Explain how you determined this price. You must use calculus techniques in solving for this price. What is the maximum possible seasonal profit? How can you be certain that the profit is maximized? You must use calculus techniques in solving for this price and how you can be certain that the profit is maximized. Explain what this value means By comparing the price you obtained for maximum revenue with the price you obtained for maximum profit, please explain any difference between the two and the reason for the difference.

Explanation / Answer

p = 70 - 0.7x

(a)

Total revenue, TR = p.x = 70x - 0.7x2

Since p & x cannot be negative, TR cannot be negative.

TR >= 0

70x - 0.7x2 >= 0

x. (70 - 0.7x) >= 0

x >= 0, and/or (70 - 0.7x) >= 0

x >= 0, and/or 70 >= 0.7x

x >= 0, and/or 100 >= x

Domain of x: 0 <= x <= 100

(b)

Revenue is highest when dTR / dx = 0

70 - 1.4x = 0

1.4x = 70

x = 70 / 1.4 = 50

p = 70 - (0.7 x 50) = 70 - 35 = 35

(c)

Marginal revenue, MR = dTR / dx = 70 - 1.4x

When x = 20, MR = 70 - (1.4 x 20) = 70 - 28 = 42

This means that, when output increases from 19 to 20 units, total revenue increases by 42 units.

(d)

C = 100 + 11.25x

Average cost (AC) = C / x = (100 / x) + 11.25

Marginal cost (MC) = dC / dx = 11.25

AC measures the average cost of producing one unit of output, but MC measures the increase in cost when one more unit of output is produced.

When x = 20,

C = 100 + (11.25 x 20) = 100 + 225 = 325

AC = C / x = 325 / 20 = 16.25 per unit

MC = 11.25

So, AC of producing one units is 16.25, but each additional unit of output increases total cost by 11.25. The difference is due to existence of fixed cost of 100.

NOTE: First 4 sub-questions are answered.

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