The given quadratic equation for the daily production cost $y$ is:

$y = 650 - 15x + 0.45x^2$

Where $x$ is the number of canned items. To find the minimum daily production cost, we need to determine the value of $x$ that minimizes the quadratic function.

Step 1: Identify the coefficients of the quadratic equation

The quadratic equation is in the standard form:

$y = ax^2 + bx + c$

Where:

• $a = 0.45$ (the coefficient of $x^2$)
• $b = -15$ (the coefficient of $x$)
• $c = 650$ (the constant)

Step 2: Find the vertex

The minimum (or maximum) value of a quadratic function occurs at the vertex. The formula to find the value of $x$ at the vertex is:

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

Substituting $a = 0.45$ and $b = -15$:

$x = \frac{-(-15)}{2(0.45)} = \frac{15}{0.9} = 16.67$

Thus, the number of canned items $x$ that minimizes the daily production cost is approximately $16.67$.

Step 3: Calculate the minimum daily production cost

Now, substitute $x = 16.67$ back into the original equation to find $y$:

$y = 650 - 15(16.67) + 0.45(16.67)^2$

First, calculate each term:

• $-15(16.67) = -250.05$
• $0.45(16.67)^2 = 0.45 \times 278.89 = 125.50$

Now, substitute these values into the equation:

$y = 650 - 250.05 + 125.50 = 525.45$

Step 4: Conclusion

The minimum daily production cost is approximately $525.00, which matches the option $525.00.

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Would you like more details on any of these steps? Here are five related questions:

1. How do we determine if a quadratic equation has a minimum or maximum?
2. Can you explain how the vertex formula works for quadratic functions?
3. How can we approximate the number of canned items for a given production cost?
4. What are some other methods to solve quadratic equations?
5. What is the significance of the coefficient of $x^2$ in quadratic functions?

Tip: The sign of the coefficient of $x^2$ determines whether the parabola opens upward (minimum) or downward (maximum).