Question

A small kayak design company has determined that when x hundred kayaks of a certain design are built,

the average cost per kayak is given by C(x)=0.57x2-0.75x+1.86, where C(x) is in hundreds of dollars.
Repeat the same procedures as in Problem 1 to find the points of interest for this new equation.
Note: Press [zoom] and select ZStandard to set the window up correctly for this problem.
7. How many kayaks should the shop build to minimize the average cost per kayak?
8. What is the cost per kayak in the minimized cost situation?

Answer 1

Therefore, the small kayak design company should build approximately 0.658 hundred kayaks to minimize the average cost per kayak. The cost per kayak in this minimized cost situation is approximately $261.20.

Explanation:

To find the minimum average cost per kayak, we need to find the value of x that minimizes the function C(x) = 0.57x^2 - 0.75x + 1.86. This can be done by finding the vertex of the quadratic function.

Finding the Vertex:

The vertex of a quadratic function in the form f(x) = ax^2 + bx + c is given by the formula:

x = -b / (2a)

In our case, a = 0.57 and b = -0.75. Plugging these values into the formula, we can find the x-coordinate of the vertex:

x = -(-0.75) / (2 * 0.57)

x = 0.75 / 1.14

x ≈ 0.658

Minimum Average Cost:

To find the minimum average cost per kayak, we substitute the value of x back into the original function:

C(x) = 0.57x^2 - 0.75x + 1.86

C(0.658) ≈ 0.57(0.658)^2 - 0.75(0.658) + 1.86

C(0.658) ≈ 0.57(0.432) - 0.494 + 1.86

C(0.658) ≈ 0.246 - 0.494 + 1.86

C(0.658) ≈ 2.612