OP Malhotra Class 12 Maths Solutions Chapter 26 Application of Calculus in Commerce and Economics Ex 26(b)

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S Chand Class 12 ICSE Maths Solutions Chapter 26 Application of Calculus in Commerce and Economics Ex 26(b)

Question 1.
Find (a) the averages and (b) the marginal cost given that for x units of commodity.
(i) C(x) = \(\frac { 1 }{ 3 }\)x³ + x² – 8x + 5;
(ii) C(x) = \(\frac { 1 }{ 3 }\)x³ + 3x³ + 3x² – 16x + 2;
(iii) \(\frac { 1 }{ 3 }\)x³ + 3x² – 7x + 16
Solution:
(i) Given C(x) = \(\frac{x^3}{3}\) + x² – 7x + 16
(a) Marginal cost function (MR) = \(\frac{d}{d x}\)C(x) = \(\frac{3 x^2}{3}\) + 2x – 8 = x² + 2x – 8
(b) Average cost function (AC) = \(\frac{\mathrm{C}(x)}{x}\) = \(\frac{x^2}{3}\) + x – 8 + \(\frac{5}{x}\)
(c) Slope of average cost function = \(\frac { d }{ dx }\)(AC) = \(\frac{d}{d x}\)\(\left[\frac{x^2}{3}+x-8+\frac{5}{x}\right]\) = \(\frac{2 x}{3}\) + 1 – \(\frac{5}{x^2}\)

(ii) Given total cost function = C(x) = \(\frac{x^3}{3}\) + 3x² – 16x + 2
Marginal cost function (MC) = \(\frac{d}{d x}\) (C(x)) = \(\frac{d}{d x}\left[\frac{x^3}{3}+3 x^2-16 x+2\right]\)
= \(\frac{3 x^2}{3}\) + 6x – 16 = x² + 6x – 12

Average cost function(AC) = \(\frac{C(x)}{x}\) = \(\frac{1}{x}\)\(\left[\frac{x^3}{3}+3 x^2-16 x+2\right]\) = \(\frac{x^2}{3}\) + 3x – 16 + \(\frac{2}{x}\)

(iii) Given C(x) = \(\frac{x^3}{3}\) + 3x² – 7x + 16
∴ average cost = \(\frac{\text { Total C }(x)}{x}\) = \(\frac{\frac{x^3}{3}+3 x^2-7 x+16}{x}\) = \(=\frac{x^2}{3}\) + 3x – 7 + \(\frac{16}{x}\)
∴ Marginal cost = MC = \(\frac{d}{d x}\)C(x) = \(\frac{d}{d x}\)\(\left[\frac{x^3}{3}+3 x^2-7 x+16\right]\) = x² + 6x – 7

Question 2.
(i) If the total cost function for a manufacturer is given by C = \(\frac{5 x^2}{\sqrt{x^2+3}}+5000\), find the marginal cost function.
(ii) The cost of producing x units of a product is given by C = 7x + 130, show that the marginal cost is always constant.
Solution:
(i) Given cost function C(x) = \(\frac{5 x^2}{\sqrt{x^2+3}}+5000\)

(ii) Given cost function C(x) = 7x + 130
∴ MC = Marginal cost function = \(\frac{d}{d x}\)C(x)
= \(\frac{d}{d x}\) (7x + 130) = 7
which is constant.

Question 3.
(i) The cost function C(x) of a function is given by C(x) = 2 x² – 4x + 5. Find the average cost, and the marginal cost when (a) x =10 (b) when x = 2.
(ii) The cost function of a firm is given by C =4x² – x + 70. Find the average cost and the marginal cost, when.
(iii) The cost function of a firm is given by C =3x² – 2x + 3. Find (i) the average cost and (ii) the marginal cost when x = 3.
Solution:
(i) Given,
C(x) = 2x² – 4x + 5
(a) Average cost (AC) = \(\frac{\mathrm{C}(x)}{x}\) = 2x – 4 + \(\frac{5}{x}\)
When x = 10; (AC)_x=10 = 2 × 10 – 4 + \(\frac{5}{10}\) = 16 + \(\frac{1}{2}\) = 16.5

(b) Marginal cost (MC) = \(\frac{d}{d x}\)C(x) = \(\frac{d}{d x}\) (2x² – 4x + 5) = 4x – 4
at x = 10; (MC)_x=10 = 4 × 10 – 4 = 36

(ii) Given cost function C = 4x² – x + 70
∴ Average cost (AC) = \(\frac{c}{x}\) = \(\frac{4 x^2-x+70}{x}\) = 4x – 1 + \(\frac{70}{x}\)
and Marginal cost (MC) = \(\frac{dc}{dx}\) = 8x – 1
∴ (AC)_{x = 3} = 12 – 1 = \(\frac{70}{3}\) = \(\frac{103}{3}\) and (MC)_{x = 3} = 8 × 3 – 1 = 23

(iii) Given cost function C = 3x² – 2x + 3
∴ average cost (AC) = \(\frac{C}{x}\) = \(\frac{3 x^2-2 x+3}{x}\) = 3x – 2 + \(\frac{3}{x}\)
Thus (AC)_{x = 3} = 9 – 2 + \(\frac{3}{3}\) = 8
∴ Marginal cost (MC) = \(\frac{dC}{dx}\) = 6x – 2
∴ (MC)_{x = 3} = 18 – 2 = 16

Question 4.
The total cost C(x), associated with production and making of x units of an item is given by C(x) = 0.005x³ – 0.02x² + 30x + 5000; find
(i) the average cost function,
(ii) the average cost of output of 10 units,
(iii) the marginal cost function, and
(iv) the marginal cost when 3 units are produced.
Solution:
Given C(x) = 0.005 x³ – 0.02x² + 30x + 5000
(i) verage cost function (AC) = \(\frac{\mathrm{C}(x)}{x}\) = \(\frac{0.005 x^3-0.02 x^2+30 x+5000}{x}\) = 0.005x² = 0.02x + 30 + \(\frac{5000}{x}\)

(ii) ∴average cost of output of 10 units = (AC)_{x = 10}
= 0.005(10)² – 0.02 × 10 + 30 + \(\frac{5000}{10}\)
= 0.5 – 0.2 + 30 + 500 = 530.3

(iii) Marginal cost function (MC) = \(\frac{dC}{dx}\) = 0.015x² – 0.04x + 30

(iv) ∴ Marginal cost when 3 units are produced = (MC)_{x = 3}
= 0.015 × 9 – 0.04 × 3 + 30
= 0.135 – 0.12 + 30 = 30.015

Question 5.
(i) If a manufacturer’s total cost function C is given by C = \(\frac{x^2}{25}+2 x\), find (a) the average cost function, (b) the marginal cost function, and (c) the marginal cost when 5 units are produced. Also, interpret the result.
(ii) The average cost function for a product is given by AC = 0.0002x³ – 0.05x + 7 + \(\frac{8000}{x}\), where x is the output. Find the marginal cost function. What is the marginal cost when 100 units are produced. Interpret your result.
Solution:
(i) Given cost function C = \(\frac{x^2}{25}\) + 2x
(a) Average cost function (AC) = \(\frac{C}{x}\) =\(\frac{1}{x}\) \(\left[\frac{x^2}{25}+2 x\right]\) = \(\frac{x}{25}\) + 2
(b) Marginal cost function (MC) = \(\frac{dC}{dx}\) = \(\frac{2x}{25}\) + 2
(c) Marginal cost when 5 units are produced = (MC)_{x = 5}
= \(\frac{2 \times 5}{25}\) + 2 = \(\frac{2}{5}\) + 2 = \(\frac{12}{5}\) = 2.4
This shows that for an increase in production by 1 unit i.e. from 5 units to 6 units, the cost of additional unit is approximately 2.4 .

(ii) Given average cost function AC = 0.0002x³ – 0.05x + 7 + \(\frac{8000}{x}\)
since AC = \(\frac{\mathrm{C}(x)}{x}\) ⇒ C(x) = AC × x = 0.0002x⁴ – 0.05x² + 7x + 8000
∴ Marginal cost function (MC) = \(\frac{dC}{dx}\) = 0.0008 x³ – 0.1x + 7
∴ (MC)_{x = 100} = 0.0008(100)³ – 0.1 × 100 + 7 = 8 × 10⁴ × 10⁶ – 10 + 7 = 797
This shows that for an increase in production by 1 unit i.e. from 100 units to 101 units, the cost of additional unit is approximately 797 .

Question 6.
(i) The total cost function for a production and marketing activity is given by
C(x) = \(\frac { 3 }{ 4 }\)x² – 7x + 27
Find the level of output (number of units produced) for which MC = AC.
(ii) If C(x) = 0.05x – 0.2x² – 5, find the level of output x for which the average cost function AC becomes equal to the marginal cost.
Solution:
(i) Given total cost function C(x) = \(\frac { 3 }{ 4 }\)x² – 7x + 27
∴Marginal cost function = MC = \(\frac { dC }{ dx }\) = \(\frac { 3 }{ 2 }\)x – 7
and Average cost function = AC = \(\frac{\mathrm{C}(x)}{x}\) = \(\frac{3}{4}\)x – 7 + \(\frac{27}{x}\)

since x > 0 ∴ x = 6
Hence the required no. of units produced be 6 for which MC = AC.

(ii) Given C(x) = 0.05x – 0.2x² – 5
∴ average cost function = AC = \(\frac { C }{ x }\) = 1 + 2x² – 3.5x
and Marginal cost function = MC =\(\frac { d }{ x }\)C(x) = 0.05 – 0.4x
We want to find the level of output x for which AC = MC
⇒ 0.05 – 0.2x – \(\frac { 5 }{ x }\) = 0.05 – 0.4x ⇒ 0.2x = \(\frac { 5 }{ x }\) ⇒ x² = \(\frac { 5 }{ 0.2}\) = 25 ⇒ x = ± 5
but x > 0 ∴ x = 5

Question 7.
The total cost function is given by C = x + 2x³ – 3.5x², find the marginal average cost function (MAC). Also, find the points where the MC curve cuts the x-axis and y-axis.
Solution:
Given total cost function C = x + 2x³ – 3.5x²
∴ average cost function AC = \(\frac { C }{ x }\) = 1 + 2x² = 3.5x
and Marginal average cost function (MAC) = \(\frac { d }{ dx }\)(AC) = 4x – 3.5
∴ MC = \(\frac { dC }{ dx }\) = 1 + 6x² – 7x

Now MC curve cuts x-axis i.e. y = θi.e. MC = 0
⇒ 6x² – 7x + 1 = 0 ⇒ (x – 1)(6x – 1) = 0 ⇒ x = 1, \(\frac { 1 }{ 6 }\)
Thus MC curve meets x-axis at (1,0) and (\(\frac { 1 }{ 6 }\), 0)
and MC curve cuts y-axis i.e. x = 0
∴ from (1); MC = 1 i.e. at point (0, 1) .

Question 8.
Verify that \(\frac { d }{ dx }\) (AC) = \(\frac { 1 }{ x }\)(MC – AC), given that
(i) C = 3 – 2x + 5x²
(ii) C= a + bx + cx²
(iii) C(x) = ax + bx² + ex + d.
Solution:
(i) Given C = 3 – 2x + 5x²


Question 9.
The average cost of producing x units of a commodity is given by the equation
AC = \(\frac{x^2}{200}\) – \(\frac{x}{50}\) – 30 + \(\frac{5000}{x}\)
Find the marginal cost (MC) function and verify that
\(\frac{d A C}{d x}\) = \(\frac{M C-A C}{x}\)
Solution:
Given average cost function AC = \(\frac{x^2}{500}\) – \(\frac{x}{50}\) – 30 + \(\frac{5000}{x}\)
since AC = \(\frac{C(x)}{x}\) ⇒ C(x) = AC × x
∴ Total cost function C(x) = \(\frac{x^3}{500}\) – \(\frac{x^2}{50}\) – 30x + 5000
Thus Marginal cost function MC = \(\frac{d \mathrm{C}}{d x}\) = \(\frac{3 x^2}{500}\) – \(\frac{2 x}{50}\) – 30

Question 10.
If C = \(2 x\left(\frac{x+4}{x+1}\right)+6\) is the total cost of production of x units of a certain product, show that the marginal cost falls continuously as the output x increases.
Solution:
Given cost function C(x) = \(2 x\left(\frac{x+4}{x+1}\right)+6\)

To show that MC falls continuously with increase in $x$, we should show that
\(\frac{d}{d x}\)(MC) < 0 ∀ x
\(\frac{d}{d x}\)(MC) = \(\frac{-12}{(x+1)^3}\) < 0 for all positive values of x. Thus, the marginal cost falls continuously as the output increases.

Question 11. Verify for the cost functions C(x) = ax \(\left(\frac{x+b}{x+c}\right)\) + d, a, b, c, d > 0, b > c that the average and (x + c) marginal cost curves fall continuously with increasing output.
Solution:
Given cost function C(x) = \(a x\left(\frac{x+b}{x+c}\right)+d\)
∴ Average cost function AC = \(\frac{C(x)}{x}\) = a \(\left(\frac{x+b}{x+c}\right)\) + \(\frac{d}{x}\)

Thus average cost function fall continuously as output x increases.

Thus Marginal cost curve fall continuously with increasing output.

Question 12.
The average cost function associated w ith producing and m arketing x units of an item is given 50 by AC = 2x – 11 + \(\frac{50}{x}\). Find the total cost function and marginal cost function. Also, find the range of the output for which AC is increasing.
Solution:
Let C(x) be the required total cost function.
Given AC = 2x – 11 + \(\frac{50}{x}\)
since AC = \(\frac{C(x)}{x}\) ⇒ C(x) = AC × x = 2x² – 11x + 50
∴ Marginal cost function (MC) = \(\frac{d}{d x}\)C(x) = \(\frac{d}{d x}\)(2x² – 11x + 50) = 4x – 11
For AC is to be increasing; \(\frac{d}{dx}\) (AC) > 0 ⇒ 4x – 11 > 0 ⇒ x > \(\frac{11}{4}\)
Thus average cost increases if the output x > \(\frac{11}{4}\)

Question 13.
The total cost function for x units is given by C(x) = \(\sqrt{6 x+5}+2500\). Show that the marginal cost decreases as the output x increases.
Solution:

Thus clearly marginal cost function decreases as the output x increases.