Advertisements
Advertisements
Question
A monopolist has a demand curve x = 106 – 2p and average cost curve AC = 5 + `x/50`, where p is the price per unit output and x is the number of units of output. If the total revenue is R = px, determine the most profitable output and the maximum profit.
Advertisements
Solution
x = 106 – 2p
(or) 2p = 106 – x
p = `1/2`(106 – x)
Revenue, R = px
= `1/2`(106 – x) x
= 53x – `x^2/2`
Average Cost, AC = `5 + x/50`
Cost C = (AC)x
= `(5 + x/50)x`
= `5x + x^2/50`
Profit (P) = Revenue – Cost
`"dP"/"dx" = 48 - (13(2x))/25`
`"dP"/"dx"` = 0 gives
`48 - (13(2x))/25` = 0
`48 = (13 xx 2x)/25`
x = `(48 xx 25)/(13 xx 2)`= 46.1538 = 46 (approximately)
Also `("d"^2"P")/"dx"^2 = 0 - (13)^2/25`, negative since `("d"^2"P")/"dx"^2` is negative, profit is maximum at x = 46 units.
Profit = `48x – 13/25` x2
When x = 46,
Profit = `48 × 46 - 13/25` × 46 × 46
`= 2208 - 27508/25`
= 2208 – 1100.32
= ₹ 1107.68
APPEARS IN
RELATED QUESTIONS
A firm wants to maximize its profit. The total cost function is C = 370Q + 550 and revenue is R = 730Q-3Q2. Find the output for which profit is maximum and also find the profit amount at this output.
Evaluate : `int_1^2 1/((x+1)(x+3)) dx`
In a firm the cost function for output x is given as C = `"x"^3/3 - 20"x"^2 + 70 "x"`. Find the 3 output for which marginal cost (Cm) is minimum.
Solve the following assignment problem to minimize the cost:
| Persons | Jobs | ||
| I | II | III | |
| A | 7 | 3 | 5 |
| B | 2 | 7 | 4 |
| C | 6 | 5 | 3 |
| D | 3 | 4 | 7 |
A manufacturer can sell x items at a price of ₹ (280 - x) each .The cost of producing items is ₹ (x2 + 40x + 35) Find the number of items to be sold so that the manufacturer can make maximum profit.
Cost of assembling x wallclocks is `( x^3/3 - 40x^2)` and labour charges are 500x. Find the number of wall clocks to be manufactured for which average cost and marginal cost attain their respective minimum.
Find the value of x for which the function `f(x) = x^3 - 3x^2 - 9x + 25` is increasing.
The total cost function y for x units is given by y = `4x((x+2)/(x+1)) + 6`. Prove that marginal cost [MC] decreases as x increases.
For the cost function C = 2000 + 1800x - 75x2 + x3 find when the total cost (C) is increasing and when it is decreasing.
The maximum value of f(x) = sin x is:
