EXERCISE -1
1) A manufacturer finds that the production cost of each article produced by his firm is ₹25 and the other fixed costs are ₹25000. If each article is sold for ₹35, Find
A) cost function. C(x)= 25000+ 25x
B) Revenue function. R(x)= 35x
C) break even point. 2500
2) The fixed cost of a new product is ₹35000 and the variable cost per unit ₹500. If the demand function p(x)= 5000 - 10x, find the break-even values. 10, 35
3) The fixed cost in the variable cost of x units of a product of a company are ₹ 30000 and ₹75x respectively. If each unit is sold for ₹125, find break-even point. 600
4) A company decides to set up a small production plant for manufacturing clocks. The total cost of initial set up(fixed cost) is ₹9 lakhs. The additional cost (variable cost) for producing each clock is ₹300. Each clock is sold at ₹750. During the first month 1500 clocks are produced and sold.
A) determine the cost function for C(x) for producing x clocks. 300x + 900000
B) determine the revenue function R(x) for the sale of x clocks. 750x
C) determine the profit function p(x) for the sale of x clocks. 450x - 900000
D) what profit or loss the company incurs during the first month when all 1500 clocks are sold ? 225000
E) determine the break even point. 2000
5) The printing cost of 1900 and 1300 copies of a book are 5100 and 3700 respectively. Find the equation of the cost curve of printing assuming it to be linear. If the selling price is ₹3 per copy, find the number of copies that must be printed so that
A) there is no profit or loss. y= 7x/3 + 2000/3, 1000 copies
B) a profit of ₹ 40. 1600 copies
6) A publishing house finds that the production cost directly attributed to each book is ₹30 and that the fixed costs are ₹15000. If each book can be sold for ₹45 then find
A) cost function. 30x+15000
B) the revenue function. 45x
C) break-even point. 1000
7) A firm knows that the demand function for one of its products is linear. It also knows that it can sell 1000 units when the price is ₹4 per u6, and it can sell 1500 units when the price is ₹2 per unit. Find
A) demand function. 8 - x/250
B) total revenue function. 8x -x²/250
C) marginal revenue function. 8 - x/125
8) ABC Co. Ltd. is planning to market a new model of shaving razor. Rather than set the selling price of the razor based only on production cost estimates, management pulls the retailer of the razors to see how many razors they would buy for various prizes. From this survey it is determined that the unit demand function (the relationship between the amount x each retailer would buy and the price p he is willing to pay) is
x= 1500 p + 30000
The fixed costs to the company for production of the razors are found to be ₹28000 and the cost for material and labour to produce each razor is estimated to be ₹8.00 per unit. What price should the company charge retailer in order to obtain a maximum profit? at 14, ₹26000, 9000
9)
EXERCISE -2
1) The unit demand function is x= (25- 2p)/3, where x is the number of units and p is the price. Let the average cost per unit be ₹40. Find
A) the revenue function R in terms of price p. (25 - 2p)/3
B) the cost function C. 40(25- 2p)/3
C) the profit function P. (-2p² + 105p - 1000)/3
D) the price per unit that maximizes the profit function. When p= 105/4
E) the maximum profit.
2) The demand function faced by a firm is p= 500 - 0.2x and its cost function is C= 25x + 10000 (p= price, x= output and C= cost). Find
A) the output at which the profits of the firm are maximum. 1187.50
B) the price it will charge. 262.50
3)
7) For a manufacturer of dry cells, the daily cost of production C for x cells is given by C(x)= ₹(2.05x + 550). If the price of a cell is ₹ 3. Determine the minimum number of cells those must be produced and Sold daily to ensure no loss. 579
9) The daily cost of production C for x units of an assembly is given by C(x)= ₹(12.5 x + 6400).
A) if each units is sold for ₹25, determine the minimum number of units that should be produced and Sold to ensure no loss. 513
B) if the selling price is reduced by ₹2.50 per unit, what would be the break-even point. 640
C) if it is known that 500 units can be sold daily, what price per unit should be charged to guarantee no loss ? 25.30
a) average cost. x²/10- 5x + 60 + 100/x
b) average variable cost. x²/10 - 5x + 60
c) marginal cost. 3x²/10- 10x+60
11) A calculator manufacturing company introduces production bonus to the workers that increases the cost of a calculator. The daily cost of production C for y calculators is given by C(y)= ₹ 2.05y + ₹ 550.
A) If each calculator is sold for ₹3, determine the minimum number that must be produced and sold daily to ensure no loss. 579
B) if the selling price is increased by 30 paise per piece, what would be the break even point ? 440
C) if it is known that at least 500 calculator can be sold daily, what price the company should charge per piece of calculator to guarantee no loss? 3.15
12) The total cost and the total revenue of a company that produces and sales x units of a product are respectively C(x)= 10x + 400 and R(x)= 60x - x³. Find
A) the break-even values. 10 & 40
B) the values of x that gives a profit. 10< x < 40
C) the values of x that results in a loss. x< 10 and x> 40
13) The total cost and the total revenue functions of a company that produces and sales x units of a particular product are given by C(x)= 5x + 350 and R(x)= 50x - x² respectively. Find
A) the break even values of x. 10 & 35
B) the values of x that produces a profit. 10< x < 35
C) the values of x that result in a loss. x< 10 and x > 35
14) The total cost and the total revenue of a company that produces and sells x units of a product are respectively C(x)= 5x +350 and R(x)= 50x - x². Find
A) the break-even values. 10 or 35
B) the value of x that produces a profit. 10< x < 35
C) the value of x that results in a loss. x< 10 or x> 35
15) The cost function C(x) of a firm is given by C(x)= 2x² - 4x + 5. Find
A) The average cost.
B) the marginal cost when x= 10. 16.5 units, 36 units
16) A firm produces x units of a article at a total cost of ₹(5+ 48/x + 3x²). Find the minimum value of the total cost. At x= 2, ₹ 41
17) A firm produces x units of output at a total cost of ₹(2x/3 + 35/2). Find the cost when the output is 4 units, the average cost of output of 10 units, and the marginal cost when output is 3 units. ₹20.16, ₹2.42, ₹0.67
18) A firm produces x units of output per week at a total cost of ₹(x³/3 - x² + 5x + 3). Find the output levels at which the marginal cost and the average variable cost attains their respective minima. 1, 1.5
19) A firm produces x tons of a valuable metal per month at a total cost C given by C= ₹(x³/3 - 5x² + 75x +10). Find at what level of output the marginal cost attains it's minimum. 5 tons.
20) Let the cost function of a firm be given by the equation: C(x)= 300x - 10x² + x³/3, where C(x) stands for cost function and x for output. Calculate the output at which
A) the marginal cost is minimum. 10
B) the average cost is minimum. 15
C) average cost is equals to Marginal cost. 15
21) The efficiency E of a small manufacturing concern depends on the number of workers w and is given by 10E = - w³/40 + 39w - 392. Find the strength of the workers which gives maximum efficiency. 20
22) A company after examining its cost structure and revenue structure has determined that the following functions approximately describe its cost and revenues:
C= 100 + 0.015x² and R= 2x where C= total cost, R= total revenue and x= number of units produced and Sold. Find the output rate which will maximum profits for the firm. 66.67
23) A firm can sell x items per week at a price p= (300 - 2x) rupees. Producing items cost the firm y rupees where y= 2x + 1000. How much production will yield maximum profits ? 74
24) The total revenue function and the total cost function of a company are given by R= 21q - q² and C= q³/3 - 3q² - 7q + 16 respectively, where q is the output of the company. Find the output at which the total revenue is maximum and the output at which the total cost is minimum. 10.5, 7
25) The demand function of a firm is p= 500 - 0.2x and its cost function is c= 25x - 10000, where p is the price and x is the output. Find the output at which the profit of the firm will maximum. Also find the price it will charge. 1187.5, 262.5
26) The demand function of a producer is 3q= 98 - 4p and its average cost is 3q +2, where q is the output and p is the price. Find the maximum profit of the producer. 33.75 units
27) A radio manufacturer finds that he can sell x radio per week at ₹p each, where p= 2(100 - x/4). His cost of production of x radios per week is ₹ (120x + x²/2). Show that his profit is maximum when the production is 40 radios per week. Find also his maximum profit per week. 1600
28) A manufacturer produces x units per month at a total cost of ₹(x²/25 + 3x + 100). There is no competition in the market and the demand follows the rule x= 75 - 3p, where p is the selling price per article. Find x such that the net revenue is maximum,. also find the monopoly price. 30, 15
29) A firm produces x units of output at a total cost of ₹(300x - 10x² + x³/3). Find
A) output at which marginal cost is minimum. 10
B) output at which average cost is minimum. 15
C) output at which average cost is equal to marginal cost. 0, 15
30) The demand function of a monopolist is given by p= 1500 - 2x - x². Find the marginal revenue for any level of output x. Also, find marginal revenue when x= 0. 1160
31) A firm produces x tons of output per week of a total cost of ₹(x³/8 - 4x² + 12x +3). Find the level of output at which average variable cost attains minimum value. 16
32) The manufacturing cost of an item consists of ₹900 as overheads, the material cost is ₹3.00 per item and the labour cost is ₹ x²/100 for x items produced. How many items must be produced to have average cost minimum. 300
33) The total cost function of a firm is C= x³/3 - 5x² + 28x +10. Where C is total cost and x is output. A tax at the rate of ₹2 per unit of output is imposed and the producer adds it to his cost. If the market demand function is given by
p= 2530 - 5x.
Where ₹ p is the price per unit of output, find the profit maximizing output and price. 50, 2280
34) Given the demand and cost functions:
p= 10 - 4x
C= 4x
Find
A) the maximum quantity, price and the profit on this level. 12
B) what will be the new equilibrium after a tax of ₹0.50 is imposed ? 12.25
*C) the tax rate that will maximizing tax revenue and determine that tax revenue. 8
35)
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