Q14) Nike develops specs for a new football helmet. They don't know how many they are going to sell, so they asked for 3 pricing options and assume the relationship is linear (or best fit).

 	Units	Cost Quote
Min Order	1,000	$3,000
Med Order	2,000	$10,000
Large Order	10,000	$45,500
What would you expect it to cost if you produced 3500 helmets? Round to nearest dollar

  $14,500 
  $15,656 

We can approach this problem by assuming a linear relationship between the number of helmets ordered and the cost, and then finding the equation of the line that best fits the given data points.

### Step 1: Define the data points.
We are given the following data points for units and costs:

- Point 1: (1,000, $3,000)
- Point 2: (2,000, $10,000)
- Point 3: (10,000, $45,500)

We need to find the linear equation in the form $ y = mx + b $, where:
- $ y $ is the cost,
- $ x $ is the number of units,
- $ m $ is the slope, and
- $ b $ is the y-intercept.

### Step 2: Calculate the slope $ m $.
To find the slope between two points, we use the formula:

\[
m = \frac{y_2 - y_1}{x_2 - x_1}
\]

Let’s first calculate the slope between Point 1 and Point 2:

\[
m_1 = \frac{10,000 - 3,000}{2,000 - 1,000} = \frac{7,000}{1,000} = 7
\]

Now, let's calculate the slope between Point 2 and Point 3:

\[
m_2 = \frac{45,500 - 10,000}{10,000 - 2,000} = \frac{35,500}{8,000} = 4.4375
\]

Since these slopes are different, the relationship is not perfectly linear, but we'll average them for simplicity:

\[
m = \frac{7 + 4.4375}{2} = 5.71875
\]

### Step 3: Find the y-intercept $ b $.
Using the slope $ m = 5.71875 $ and one of the points, say (1,000, 3,000), we can substitute into the equation $ y = mx + b $ to solve for $ b $:

\[
3,000 = 5.71875(1,000) + b
\]

\[
3,000 = 5,718.75 + b
\]

\[
b = 3,000 - 5,718.75 = -2,718.75
\]

So, the equation of the line is:

\[
y = 5.71875x - 2,718.75
\]

### Step 4: Calculate the cost for 3,500 helmets.
Now substitute $ x = 3,500 $ into the equation:

\[
y = 5.71875(3,500) - 2,718.75
\]

\[
y = 20,015.625 - 2,718.75 = 17,296.875
\]

Rounding to the nearest dollar, the estimated cost is **$17,297**.

### Conclusion:
Based on this linear estimation, the cost for producing 3,500 helmets would be approximately **$17,297**, which does not match either $14,500 or $15,656 exactly.

Would you like further details on how this estimate was derived?

Here are some related questions to explore:
1. What if the relationship between cost and units is not linear?
2. How would using a quadratic model change the estimate?
3. Could this pricing strategy be optimized for different quantities?
4. What impact does bulk production have on cost efficiency?
5. How do companies typically calculate break-even points for production?

**Tip:** For better accuracy, use piecewise linear models or non-linear regression when fitting real-world data that doesn't align perfectly with a single trend.

Nike develops specs for a new football helmet. They don't know how many they are going to sell, so they asked for 3 pricing options and assume the relationship is linear (or best fit). 

Units	Cost Quote
Min Order	1,000	$3,000
Med Order	2,000	$10,000
Large Order	10,000	$45,500
What would you expect it to cost if you produced 3500 helmets? Round to nearest dollar

 $14,500 
 $15,656

Learn how to estimate the cost of producing 3,500 football helmets using a linear approximation based on pricing options for 1,000, 2,000, and 10,000 units. This step-by-step guide covers key algebraic concepts such as calculating slopes and solving linear equations. Suitable for Grades 9-12.

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