The total daily cost (in dollars) of producing \( x \) park benches is given by \( C(x)=3125+15 x+0.2 x^{2} \) A. Sketch the graphs of the average cost function and the marginal cost function on the same set of coordinale axes. B. Find the minimum average cost
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To sketch the graphs of the average cost function and the marginal cost function, you first need to find the average cost \( A(x) \) and the marginal cost \( C'(x) \). The average cost function is calculated as: \[ A(x) = \frac{C(x)}{x} = \frac{3125 + 15x + 0.2x^2}{x} = \frac{3125}{x} + 15 + 0.2x \] The marginal cost function is the derivative of the total cost function: \[ C'(x) = \frac{d}{dx}(3125 + 15x + 0.2x^2) = 15 + 0.4x \] Now you can graph \( A(x) \) and \( C'(x) \) on the same coordinate axes, typically with \( x \) (number of benches) on the horizontal axis and cost on the vertical axis. To find the minimum average cost, you would set \( A'(x) = 0 \): \[ A'(x) = -\frac{3125}{x^2} + 0.2 = 0 \] Solving this gives \( x^2 = 15625 \), so \( x = 125 \). To find the minimum average cost, substitute \( x = 125 \) back into \( A(x) \): \[ A(125) = \frac{3125}{125} + 15 + 0.2(125) = 25 + 15 + 25 = 65 \] Thus, the minimum average cost is $65.
