Question

The cost C (in dollars) for ordering and storing x units is C= 2x+ 500,000/x. What order size will produce a minimum cost? Use the calculus technique of your choice, but show each step. Roudn answer to nearest whole number

Answer 1

To find the order size that will produce a minimum cost, differentiate the cost function with respect to x and set the derivative equal to zero. Solve for x to find the critical point, which represents the order size that will minimize the cost. The order size is approximately 500 units.

Step-by-step explanation:

To find the order size that will produce a minimum cost, we need to find the minimum point on the graph of the cost function. Let's differentiate the cost function with respect to x to find the critical points.

C = 2x + 500,000/x

C' = 2 - 500,000/x²

Setting C' = 0, we get:

2 - 500,000/x² = 0

500,000/x² = 2

x^2 = 500,000/2

x^2 = 250,000

x = sqrt(250,000)

x ≈ 500

Rounding to the nearest whole number, the order size that will produce a minimum cost is approximately 500 units.

Answer 2

To find the order size that will produce a minimum cost, we need to minimize the cost function C(x) = 2x + 500,000/x. Using the quotient rule and setting the derivative equal to 0, we find that an order size of 500 units will produce the minimum cost.

Step-by-step explanation:

To find the order size that will produce a minimum cost, we need to minimize the cost function C(x) = 2x + 500,000/x. We can do this using calculus. Since the cost function has a rational expression in it, we can use the quotient rule to find the derivative. Taking the derivative of C(x) and setting it equal to 0 will give us the critical points. We can then determine which critical point corresponds to a minimum value.

1. Find the derivative of C(x) using the quotient rule: C'(x) = (2 - 500,000/x^2)
2. Set the derivative equal to 0 and solve for x: (2 - 500,000/x²) = 0
3. Find the critical points by solving the equation: x² = 250,000
4. Take the positive square root of 250,000 to get: x = 500

Therefore, an order size of 500 units will produce the minimum cost.