Problem 87 - Algebra and Trigonometry Real Mathematics, Real People

Question

The ordering and transportation cost \(C\) (in thousands of dollars) for the components used in manufacturing a product is given by $$C=100\left(\frac{200}{x^{2}}+\frac{x}{x+30}\right), \quad x \geq 1$$ where \(x\) is the order size (in hundreds). Use a graphing utility to graph the cost function. From the graph, estimate the order size that minimizes cost.

Step-by-Step Solution

Answer

The order size that minimizes cost can be estimated from the graph. The precise value might depend on the resolution of the tool used for graphing.

Step 1: Understand the cost function

The function given is \(C=100\left(\frac{200}{x^{2}}+\frac{x}{x+30}\right)\), where \(C\) is the ordering and transportation cost in thousands of dollars and \(x\) is the order size in hundreds. Our primary task is to find the minimum point of this function using a graphing tool.

Step 2: Plotting the cost function

To visualize the behavior of the cost function, you must plot it on your graphing tool. The plot should depict the order size \(x\) on the x-axis and the cost \(C\) on the y-axis. This graph should give a clear representation of how the cost varies with the order size and, more importantly, an approximate location of where the minimum cost occurs.

Step 3: Estimate the minimizing order size

From the graph plotted in the previous step, find the low point of the curve, or in other words, the order size at which the cost is minimized. This can be achieved by finding the x-coordinate of the lowest point on the curve. Do note that this is an approximation and the exact value may differ.