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.
*
5 points
$1025.00
$1,186.25
$536.25
$525.00

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**.

---

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).

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?

Explore how to find the minimum daily production cost at a canning company using a quadratic equation y = 650 - 15x + 0.45x^2. Learn the steps of solving for the vertex of a quadratic function to find the minimum cost. Ideal for Grades 9-11 students.

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