Questions: The Student Government Association is making Mother's Day gift baskets to sell at a fund-raiser, If the SGA makes a larger quantity of baskets, it can purchase materials in bulk. The total cost (in hundreds of dollars) of making x gift baskets can be approximated by C(x) = (6x + 1) / (x + 60). Complete parts (a) through (c) below. (a) Find the marginal cost function and the marginal cost at x = 20 and x = 40.

The Student Government Association is making Mother's Day gift baskets to sell at a fund-raiser, If the SGA makes a larger quantity of baskets, it can purchase materials in bulk. The total cost (in hundreds of dollars) of making x gift baskets can be approximated by C(x) = (6x + 1) / (x + 60). Complete parts (a) through (c) below.
(a) Find the marginal cost function and the marginal cost at x = 20 and x = 40.
Transcript text: The Student Government Association is making Mother's Day gift baskets to sell at a fund-raiser, If the SGA makes a larger quantity of baskets, it can purchase materials in bulk. The total cost (in hundreds of dollars) of making $x$ gift baskets can be approximated by $C(x)=\frac{6 x+1}{x+60}$. Complete parts (a) through (c) below. (a) Find the marginal cost function and the marginal cost at $\mathrm{x}=20$ and $\mathrm{x}=40$.
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Solution

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Solution Steps

To find the marginal cost function, we need to compute the derivative of the cost function C(x) C(x) . The marginal cost at a specific number of baskets, say x=20 x = 20 and x=40 x = 40 , is then found by evaluating this derivative at those points.

Step 1: Find the Marginal Cost Function

The marginal cost function is obtained by differentiating the total cost function C(x)=6x+1x+60 C(x) = \frac{6x + 1}{x + 60} . 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}

Step 2: Evaluate the Marginal Cost at x=20 x = 20

To find the marginal cost when x=20 x = 20 , we substitute x=20 x = 20 into the marginal cost function:

C(20)=3596400 C'(20) = \frac{359}{6400}

Step 3: Evaluate the Marginal Cost at x=40 x = 40

Similarly, to find the marginal cost when x=40 x = 40 , we substitute x=40 x = 40 into the marginal cost function:

C(40)=35910000 C'(40) = \frac{359}{10000}

Final Answer

The marginal cost function is

C(x)=6x+60(6x+1)(x+60)2 C'(x) = \frac{6}{x + 60} - \frac{(6x + 1)}{(x + 60)^2}

The marginal cost at x=20 x = 20 is

3596400 \boxed{\frac{359}{6400}}

The marginal cost at x=40 x = 40 is

35910000 \boxed{\frac{359}{10000}}

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