Manufacturing cost The manager of a furniture factory finds...

Manufacturing cost The manager of a furniture factory finds that it costs 2200 dollar to produce 100 chairs in one day and 4800 dollar to produce 300 chairs in one day. (a) Assuming that the relationship between cost and the number of chairs produced is linear, find a linear function $C$ that models the cost of producing $x$ chairs in one day. (b) Draw a graph of $C .$ What is the slope of this line? (c) At what rate does the factory's cost increase for every additional chair produced?

Manufacturing cost The manager of a furniture factory finds that it costs 2200 dollar to produce 100 chairs in one day and 4800 dollar to produce 300 chairs in one day.
(a) Assuming that the relationship between cost and the number of chairs produced is linear, find a linear function $C$ that models the cost of producing $x$ chairs in one day.
(b) Draw a graph of $C .$ What is the slope of this line?
(c) At what rate does the factory's cost increase for every additional chair produced?

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Key Concepts

Linear Functions

A linear function is an equation that creates a straight-line graph when plotted on a coordinate plane. Its standard form is C(x) = mx + b, where 'm' represents the slope (or the rate of change) and 'b' represents the y-intercept (or the fixed cost in a manufacturing context). This concept is used to model scenarios where the change between variables, such as cost and production quantity, is consistent.

Slope (Rate of Change)

The slope of a line in a linear function represents the rate at which the dependent variable changes with respect to the independent variable. In manufacturing cost problems, the slope indicates the additional cost incurred for producing one more unit. It is calculated as the change in cost divided by the change in the number of chairs produced, providing a clear understanding of incremental costs.

Modeling Manufacturing Costs

Modeling manufacturing costs with linear functions involves establishing a relationship between the number of items produced and the overall production cost. This method assumes a fixed cost plus a variable cost that linearly increases with production. The model simplifies analysis and decision-making regarding production scale and cost management.

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