Balbharati Maharashtra State Board12th Commerce Maths Solution Book PdfChapter 4 Applications of Derivatives Ex 4.4 Questions and Answers.
Maharashtra State Board 12th Commerce Maths Solutions Chapter 4 Applications of Derivatives Ex 4.4
Solution & Step-by-Step Answer:
D = 40 – ∴ = 0 – × 1 = Hence, the given function is decreasing function.
Solution & Step-by-Step Answer:
P = 183 + 120D – 3D2 ∴ (183 + 120D – 3D2) = 0 + 120 × 1 – 3 × 2D = 120 – 6D If price P is increasing, then > 0 ∴ 120 – 6D > 0 ∴ 120 > 6D ∴ D < 20 Hence, the price is increasing when D < 20.
Solution & Step-by-Step Answer:
The cost function is given as C = 100 + 600x – 3x2 ∴ (100 + 600x – 3x2) = 0 + 600 × 1 – 3 × 2x = 600 – 6x If the total cost is decreasing, then < 0 ∴ 600 – 6x < 0 ∴ 600 < 6x ∴ x > 100 Hence, the total cost is decreasing for x > 100.
Solution & Step-by-Step Answer:
(i) Let R be the total revenue. Then R = P.x = (240 – x)x ∴ R = 240x – x2 ∴ (240x – x2) = 240 × 1 – 2x = 240 – 2x R is increasing, if > 0 i.e. if 240 – 2x > 0 i.e. if 240 > 2x i.e. if x < 120 Hence, the revenue is increasing, if x < 120.
(ii) Profit π = R – C
∴ π = (240x – x2) – (180 + 4x)
= 240x – x2– 180 – 4x
= 236x – x2– 180
∴ (236x – x2– 180)
= 236 × 1 – 2x – 0
= 236 – 2x
Profit is increasing, if > 0
i.e. if 236 – 2x > 0
i.e. if 236 > 2x
i.e. if x < 118
Hence, the profit is increasing, if x < 118.
Solution & Step-by-Step Answer:
(i) Total cost C = labour cost + processing cost ∴ C = 150 – 54x + x2 ∴ (150 – 54x + x2) = 0 – 54 × 1 + 2x = -54 + 2x The total cost is decreasing, if < 0 i.e. if -54 + 2x < 0 i.e. if 2x < 54 i.e. if x < 27 Hence, the total cost is decreasing, if x < 27.
(ii) The total revenue R is given as
R = p.x
R = (10800 – 4x2) x
R = 10800x – 4x3
∴ (10800x – 4x3)
= 10800 × 1 – 4 × 3x2
= 10800 – 12x2
The revenue is increasing, if > 0
i.e. if 10800 – 12x2> 0
i.e. if 10800 > 12x2
i.e. if x2< 900
i.e. if x < 30 ……[∵ x > 0]
Hence, the revenue is increasing, if x < 30.
Solution & Step-by-Step Answer:
The total cost is given as C = 47x + 300x2 – x4 ∴ the average cost is given by (i) CA is increasing, if > 0 i.e. if 300 – 3x2 > 0 i.e. if 300 > 3x2 i.e. if x2 < 100 i.e. if x < 10 …..[∵ x > 0] Hence, the average cost is increasing, if x < 10.
(ii) CA is decreasing, if < 0
i.e. if 300 – 3x2< 0
i.e. if 300 < 3x2i.e. if x2> 100
i.e. if x > 10 ……[∵ x > 0]
Hence, the average cost is decreasing, if x > 10.
Solution & Step-by-Step Answer:
Given RA = 45 and η = 5 Now, Rm = = 45(1 – ) = 45() = 36 Hence, the marginal revenue = 36.
(ii) Find the price, if the marginal revenue is 28 and elasticity of demand is 3.
Solution:
Given Rm= 28 and η = 3
Hence, the price = 42.
(iii) Find the elasticity of demand, if the marginal revenue is 50 and price is ₹ 75.
Solution:
Given Rm= 50 and RA= 75
Hence, the elasticity of demand = 3.
Solution & Step-by-Step Answer:
The demand function is Elasticity of demand is given by Hence, the elasticity of demand at p = 4 is 3.6
Solution & Step-by-Step Answer:
The demand function is Elasticity of demand is given by
Solution & Step-by-Step Answer:
The demand function is D = 50 – 3p – p2 ∴ (50 – 3p – p2) = 0 – 3 × 1 – 2p = -3 – 2p Elasticity of demand is given by (i) When p = 5, then Since, η >1, the demand is elastic. (ii) When p = 2, then Since, 0 < η < 1, the demand is inelastic.
Solution & Step-by-Step Answer:
The demand function is The elasticity of demand is given by (i) When p = 10, then Since, η > 1, the demand is elastic. (ii) When p = 6, then Since, 0 < η < 1, the demand is inelastic.
Solution & Step-by-Step Answer:
(i) The total revenue R is given by R = p.x = (120 – x)x ∴ R = 120x – x2 ∴ (120x – x2) = 120 × 1 – 2x = 120 – 2x If the revenue is increasing, then > 0 ∴ 120 – 2x > 0 ∴ 120 > 2x ∴ x < 60 Hence, the revenue is increasing when x < 60.
(ii) Profit π = R – C
= (120x – x2) – (40 + 2x)
= 120x – x2– 40 – 2x
= 118x – x2– 40
∴ (118x – x2– 40)
= 118 × 1 – 2x – 0
= 118 – 2x
If the profit is increasing, then > 0
∴ 118 – 2x > 0
∴ 118 > 2x
∴ x < 59
Hence, the profit is increasing when x < 59.
(iii) p = 120 – x
∴ x = 120 – p
∴ (120 – p)
= 0 – 1
= -1
Elasticity of demand is given by
Solution & Step-by-Step Answer:
Ec = (0.0003)I2 + (0.075)I MPC = [(0.0003)I2 + (0.075)I] = (0.0003)(2I) + (0.075)(1) = (0.0006)I + 0.075 When I = 1000, then MPC = (0.0006)(1000) + 0.075 = 0.6 + 0.075 = 0.675. ∴ MPC + MPS = 1 ∴ 0.675 + MPS = 1 ∴ MPS = 1 – 0.675 = 0.325 Now, APC = = (0.0003)I + (0.075) When I = 1000, then APC = (0.0003)(1000) + 0.075 = 0.3 + 0.075 = 0.375 ∵ APC + APS = 1 ∴ 0.375 + APS = 1 ∴ APS = 1 – 0.375 = 0.625 Hence, MPC = 0.675, MPS = 0.325, APC = 0.375, APS = 0.625.