Question

The equation for the cost in dollars of producing computer chips is y =.000015x^2-.03x+35. Where x is the number of chips produced . Find the number of chips that minimizes the cost. What is the cost for that number of chips?

Answer 1

1000 chips minimize the production cost in the given equation, and the minimum cost for producing these chips is $20.

Step-by-step explanation:

To find the number of chips that minimizes the cost in the given quadratic equation, y = 0.000015x^2 - 0.03x + 35, we can use the vertex formula for a parabola, which is derived from the general quadratic equation.

The general form of a quadratic equation is y = ax^2 + bx + c, and the x-coordinate of the vertex, which gives us the number of chips that minimizes the cost, can be found using x = -b/(2a). For our equation, a = 0.000015 and b = -0.03.

Substituting the values of a and b into the vertex formula:

1. x = -(-0.03)/(2 * 0.000015)
2. x = 0.03/0.00003
3. x = 1000 (number of chips)

Now, we plug this value back into the original equation to find the minimum cost:

1. y = 0.000015(1000)^2 - 0.03(1000) + 35
2. y = 15 - 30 + 35
3. y = $20 (cost for producing 1000 chips)

This gives us the minimum cost and the number of chips that lead to this cost. Therefore, 1000 chips minimize the cost, and the minimum cost for producing these chips is $20.

Answer 2

In order to find the number of chips that would result in the minimum cost, we take the first derivative of the given equation. Note that the derivative refers to the slope of the graph at a given point. We can utilize this concept knowing that at the minimum or maximum point of a graph, the slope is zero.

Taking the derivative of the given equation and equating it to zero, we have:

y' = (0.000015)(2)x - (0.03)x° + 0
0 = (0.00003)x - 0.03

Solving for x or the number of chips produced, we have x = 1000. We then substitute this value in the given equation, such that,

y = (0.000015)(1000)² - (0.03)(1000) + 35

The minimized cost, y, to produce 1000 chips is then calculated to be $20.