To find the company's linear cost function, we need to use the information provided:

1. Fixed Costs: The fixed cost is $\$525$.
2. Variable Costs: We know that the total cost to produce 15 items is $\$1,485$.

Step 1: Determine the Variable Cost Per Item

The total cost is the sum of the fixed costs and the variable costs. The variable cost per item can be found by subtracting the fixed costs from the total cost and then dividing by the number of items produced.

$\text{Total Cost} = \text{Fixed Costs} + \text{Variable Cost per Item} \times \text{Number of Items}$

Given that the total cost for 15 items is $\$1,485$:

$1,485 = 525 + (\text{Variable Cost per Item}) \times 15$

Now, solve for the variable cost per item:

$1,485 - 525 = 15 \times (\text{Variable Cost per Item})$ $960 = 15 \times (\text{Variable Cost per Item})$ $\text{Variable Cost per Item} = \frac{960}{15} = 64$

So, the variable cost per item is $\$64$.

Step 2: Write the Linear Cost Function

The linear cost function can be expressed in slope-intercept form:

$C(x) = \text{Fixed Cost} + (\text{Variable Cost per Item}) \times x$

Substituting the values we found:

$C(x) = 525 + 64x$

Thus, the company's linear cost function is:

$\boxed{C(x) = 64x + 525}$

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Would you like to go over any details or have any questions? Here are five related questions you might find helpful:

1. How would you calculate the cost of producing 20 items using this cost function?
2. What is the significance of the slope in the cost function?
3. If the company wanted to reduce the fixed costs, how would that affect the cost function?
4. How would the cost function change if the variable cost per item increased to $70?
5. Can the company break even if the price per item is reduced to $60? If so, how many items must be sold?

Tip: In linear functions like this one, the slope represents the rate of change—in this case, the cost added for each additional item produced.