In a proposal to some clients, a housing contractor listed...

In a proposal to some clients, a housing contractor listed the following costs:
a. Write a linear function that the clients could use to determine the cost of building a home having $f$ square feet.
b. Find the cost to build a home having $1,950$ square feet.

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Key Concepts

Linear Functions

A linear function represents a relationship between two variables where the rate of change is constant. In the context of cost modeling, it is used to express how total cost depends on a variable such as area, and it can be written in the slope-intercept form (y = mx + b), where m is the slope and b is the y-intercept.

Fixed and Variable Costs

In many practical applications like building construction, costs are often divided into fixed costs (which remain constant regardless of the size or scale, such as a base fee) and variable costs (which change proportionally with the quantity, such as cost per square foot). The linear function captures both these elements, with the y-intercept representing the fixed cost and the slope representing the variable cost per unit.

Function Evaluation

Once a linear function is established, function evaluation involves substituting a specific input value into the function to calculate the corresponding output. This process is essential for determining the actual cost for a specific case, such as the cost to build a home with a given number of square feet.

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