Questions: Finding Linear Models A company manufactures small refrigerators. The total cost to manufacture 63 refrigerators is 9275. The total costs to manufacture 174 refrigerators is 24260. Assume that total cost, C, is linearly related to the number of refrigerators, x, the company manufactures and includes a fixed cost and a cost per refrigerator A) Write the cost function, C(x) in terms of x, the number of refrigerators manufactured. B) The total cost to manufacture 185 refrigerators is C) The fixed costs are D) The cost to produce each additional refrigerator is

Solution Steps

To solve this problem, we need to determine the linear cost function \( C(x) = mx + b \), where \( m \) is the cost per refrigerator and \( b \) is the fixed cost. We have two points: (63, 9275) and (174, 24260). First, calculate the slope \( m \) using these points. Then, use one of the points to solve for \( b \). Once we have the cost function, we can calculate the total cost for 185 refrigerators, identify the fixed cost, and determine the cost per additional refrigerator.

We have two points representing the total cost to manufacture refrigerators:

• \( (63, 9275) \)
• \( (174, 24260) \)

The slope \( m \) of the linear function \( C(x) = mx + b \) is calculated as follows: \[ m = \frac{C_2 - C_1}{x_2 - x_1} = \frac{24260 - 9275}{174 - 63} = 135.0 \]

Next, we find the fixed cost \( b \) using one of the points: \[ b = C_1 - m \cdot x_1 = 9275 - 135.0 \cdot 63 = 770.0 \]

Thus, the cost function is: \[ C(x) = 135.0x + 770.0 \]

Using the cost function, we can find the total cost to manufacture 185 refrigerators: \[ C(185) = 135.0 \cdot 185 + 770.0 = 25745.0 \]

From our calculations:

• The fixed costs \( b \) are \( 770.0 \).
• The cost to produce each additional refrigerator \( m \) is \( 135.0 \).

Final Answer