To find the marginal cost function, we need to compute the derivative of the cost function C(x) C(x) C(x). The marginal cost at a specific number of baskets, say x=20 x = 20 x=20 and x=40 x = 40 x=40, is then found by evaluating this derivative at those points.
The marginal cost function is obtained by differentiating the total cost function C(x)=6x+1x+60 C(x) = \frac{6x + 1}{x + 60} C(x)=x+606x+1. The derivative is given by:
C′(x)=6x+60−(6x+1)(x+60)2 C'(x) = \frac{6}{x + 60} - \frac{(6x + 1)}{(x + 60)^2} C′(x)=x+606−(x+60)2(6x+1)
To find the marginal cost when x=20 x = 20 x=20, we substitute x=20 x = 20 x=20 into the marginal cost function:
C′(20)=3596400 C'(20) = \frac{359}{6400} C′(20)=6400359
Similarly, to find the marginal cost when x=40 x = 40 x=40, we substitute x=40 x = 40 x=40 into the marginal cost function:
C′(40)=35910000 C'(40) = \frac{359}{10000} C′(40)=10000359
The marginal cost function is
The marginal cost at x=20 x = 20 x=20 is
3596400 \boxed{\frac{359}{6400}} 6400359
The marginal cost at x=40 x = 40 x=40 is
35910000 \boxed{\frac{359}{10000}} 10000359
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