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.

Solution Steps

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

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

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

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

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

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

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

Final Answer