Tamil Nadu Board of Secondary EducationHSC Commerce Class 11

# A firm produces x tonnes of output at a total cost of C(x) = 1/10x^3 - 4x^2 - 20x + 7 find the - Business Mathematics and Statistics

Sum

A firm produces x tonnes of output at a total cost of C(x) = 1/10x^3 - 4x^2 - 20x + 7 find the

1. average cost
2. average variable cost
3. average fixed cost
4. marginal cost and
5. marginal average cost.

#### Solution

c(x) = f(x) + x

c(x) = 1/10x^3 - 4x^2 - 20x + 7

Then f(x) = 1/10x^3 - 4x^2 - 20x and k = 7

(i) Average Cost (AC) = "Total cost"/"Output" = ("C"(x))/x = ("f"(x) + k)/x

= (1/10x^3 - 4x^2 - 20x + 7)/x

= 1/10 x^3/x - (4x^2)/x - (20x)/x + 7/x

= 1/10 x^2 - 4x - 20 + 7/x

(ii) Average Variable Cost (AVC) = "Variable cost"/"Output" = ("f"(x))/x

= (1/10 x^3 - 4x^2 - 20x)/x

= 1/10 x^3/x - (4x^2)/x - (20x)/x

= 1/10 x^2 - 4x - 20

(iii) Average Fixed Cost (AFC) = "Fixed cost"/"Output" = k/x = 7/x

(iv) Marginal Cost (MC) = "dC"/"dx"

= "d"/"dx"(1/10 x^3 - 4x^2 - 20x + 7)

= "d"/"dx" (1/10 x^3) - "d"/"dx"(4x^2) - "d"/"dx" (20x) + "d"/"dx" (7)

= 1/10 "d"/"dx" (x^3) - 4"d"/"dx"(x^2) - 20"d"/"dx" (x) + 0

= 1/10 (3)x^(3-1) - 4(2)^(2-1) - 20(1)

= 3/10 x^2 - 8x - 20

(v) Marginal Average Cost (MAC) = "d"/"dx"(AC)

= "d"/"dx"(1/10 x^2 - 4x - 20) + 7/x

= 1/10"d"/"dx"(x^2) - 4 "d"/"dx" (x) - "d"/"dx"(20) + "d"/"dx"(7/x)

= 1/10 (2x^(2-1)) - 4(1) - 0 - 7/x^2

= 1/5x - 4 - 7/x^2

Concept: Applications of Differentiation in Business and Economics
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Chapter 6: Applications of Differentiation - Exercise 6.1 [Page 138]