At a canning company the daily production cost, y, is given by the quadratic equation y=650-15x +  0.45x^2  , where x is the number of canned items.
What is the MINIMUM daily production cost?

Reminder x^2 same as squaring the number. Means 2 is an exponent.

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**.

---

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).

At a canning company the daily production cost, y, is given by the quadratic equation y=650-15x +  0.45x^2  , where x is the number of canned items. What is the MINIMUM daily production cost?

This problem shows how to find the minimum daily production cost for a canning company using the quadratic equation y = 650 - 15x + 0.45x^2. The solution uses the vertex formula to find the minimum value and calculates that the minimum cost is approximately $525.09. Suitable for Grades 9-12.

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