To find the minimum daily production cost, we need to determine the minimum value of the quadratic function:

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

This quadratic equation is in the standard form:

$y = ax^2 + bx + c$

Where:

• $a = 0.45$,
• $b = -15$,
• $c = 650$.

For any quadratic equation, the vertex of the parabola (which represents the minimum or maximum point) occurs at:

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

Step 1: Calculate $x$ at the vertex

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

So, the number of canned items at the minimum production cost is $x = 16.67$.

Step 2: Calculate the minimum production cost

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

$y = 650 - 15(16.67) + 0.45(16.67)^2$ $y = 650 - 250.05 + 0.45(278.09)$ $y = 650 - 250.05 + 125.14 = 525.09$

Thus, the minimum daily production cost is approximately $525.09.

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Would you like more details or have any further questions? Here are 5 related questions you might find helpful:

1. How does changing the coefficient of $x^2$ affect the shape of the graph of a quadratic function?
2. What would happen if the coefficient of $x$ were positive instead of negative?
3. How can we use calculus (derivatives) to find the minimum value of a quadratic function?
4. How does the vertex formula work for different types of quadratic equations?
5. How would the result change if the daily production cost formula included higher powers of $x$?

Tip: The sign of the coefficient of $x^2$ in a quadratic function determines whether the parabola opens upwards (minimum point) or downwards (maximum point).