To minimize the cost, we need to find the vertex of the quadratic equation $y = 0.008x^2 - 0.04x + 75$. The formula for the $x$-coordinate of the vertex for a quadratic equation in standard form $y = ax^2 + bx + c$ is:

$x = -\frac{b}{2a}$

Here:

• $a = 0.008$
• $b = -0.04$
• $c = 75$ (not used in this step)

Step 1: Find the $x$-coordinate of the vertex

$x = -\frac{-0.04}{2(0.008)} = \frac{0.04}{0.016} = 2.5$

Step 2: Interpret the result

The number of lawn mowers that should be produced to minimize the cost is 2.5. Since production must be in whole numbers, we interpret this based on the specific scenario:

• If partial production is possible, 2.5 lawn mowers minimizes cost.
• If only whole numbers are valid, you may want to calculate the cost at $x = 2$ and $x = 3$ to determine which is closer to the minimum.

Would you like a comparison for $x = 2$ and $x = 3$ to finalize the result?

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Related Questions:

1. What is the minimum cost associated with producing 2.5 lawn mowers?
2. How does rounding $x = 2.5$ to 2 or 3 affect the cost?
3. Can you explain how to find the vertex of a quadratic function in general terms?
4. What is the significance of the coefficient $a$ in this equation?
5. How does this formula apply to real-world cost optimization problems?

Tip: Quadratic functions with $a > 0$ always have a minimum point, making this vertex formula essential in cost minimization problems!