T-Shirt cost Joanne Wendelken sells silk-screened T-shirts...

T-Shirt cost Joanne Wendelken sells silk-screened T-shirts at community festivals and crafts fairs. Her marginal cost to pro- duce one T-shirt is $\$ 3.50 .$ Her total cost to produce 60 T-shirts is $\$ 300,$ and she sells them for $\$ 9$ each. (a) Find the linear cost function for Joanne's T-shirt production. (b) How many T-shirts must she produce and sell in order to break even? (c) How many T-shirts must she produce and sell to make a profit of $\$ 500 ?$

T-Shirt cost Joanne Wendelken sells silk-screened T-shirts at
community festivals and crafts fairs. Her marginal cost to pro-
duce one T-shirt is $\$ 3.50 .$ Her total cost to produce 60 T-shirts
is $\$ 300,$ and she sells them for $\$ 9$ each.
(a) Find the linear cost function for Joanne's T-shirt production.
(b) How many T-shirts must she produce and sell in order to
break even?
(c) How many T-shirts must she produce and sell to make a
profit of $\$ 500 ?$

Key Concepts

Profit Calculation

Profit calculation involves determining the amount by which total revenue exceeds total cost. In linear models, after establishing the cost function, profit is computed as the difference between revenue (price per unit times number of units) and total cost, and it helps to set targets such as making a specific profit amount.

Break-even Analysis

Break-even analysis is the process of determining the production level at which total revenues equal total costs, meaning no profit or loss is incurred. This technique is crucial for understanding how many units must be sold in order to cover all fixed and variable costs, ensuring a business's sustainability.

Linear Cost Function

A linear cost function expresses the total cost of production as a sum of fixed costs plus a constant marginal cost multiplied by the number of units produced. It takes the form C(x) = F + (marginal cost × x), where x is the quantity of units. This function is central to analyzing production costs and making decisions about scaling production.

Fixed Cost

Fixed cost involves the expenses that do not change with the level of production. It is the base cost incurred even when no units are produced. In constructing a total cost function, fixed costs are added to the variable costs to form a complete picture of production expenses.

Marginal Cost

Marginal cost refers to the cost of producing one additional unit of output. In problems involving production and cost functions, it is used to determine how the total cost changes with each unit produced, and a constant marginal cost simplifies the analysis by making the incremental cost per unit uniform.

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