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Question
If the total cost function is given by C = 5x3 + 2x2 + 1; Find the average cost and the marginal cost when x = 4.
Solution
Total cost function C = 5x3 + 2x2 + 1
Average cost = `"C"/x`
=`(5x^3 + 2x^2 + 1)/x`
= 5x2 + 2x +`1/x`
When x = 4, Average cost = 5(4)2 + 2(4) + `1/4`
= 80 + 8 + `1/4`
= `(320+32+1)/4`
= `353/4`
Marginal cost = `("dC")/("d"x)`
= `"d"/"d"x (5x^3 + 2x^2 + 1)`
=`5"d"/"d"(x^3) + 2"d"/"d"(x^2) + "d"/"d"(1)`
=5(3x2) + 2(2x) + 0
= 15x2 + 4x
When x = 4, marginal cost = `(("dC")/("d"x))_("x" = 4)`
= 15(4)2 + 4(4)
= 240 + 16
= 256
∴ The average cost and marginal cost at x = 4 are `353/4` and 256 respectively.
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