[T] A family bakery makes cupcakes and sells them at local...

[T] A family bakery makes cupcakes and sells them at local outdoor festivals. For a music festival, there is a fixed cost of $\$ 125$ to set up a cupcake stand. The owner estimates that it costs $\$ 0.75$ to make each cupcake. The owner is interested in determining the total cost $C$ as a function of number of cupcakes made. a. Find a linear function that relates cost $C$ to $x,$ the number of cupcakes made. b. Find the cost to bake 160 cupcakes. c. If the owner sells the cupcakes for $\$ 1.50$ apiece, how many cupcakes does she need to sell to start making profit? (Hint: Use the INTERSECTION function on a calculator to find this number.)

[T] A family bakery makes cupcakes and sells them at local outdoor festivals. For a music festival, there is a fixed cost of $\$ 125$ to set up a cupcake stand. The owner estimates that it costs $\$ 0.75$ to make each cupcake. The owner is interested in determining the total cost $C$ as a function of
number of cupcakes made.
a. Find a linear function that relates cost $C$ to $x,$ the number of cupcakes made.
b. Find the cost to bake 160 cupcakes.
c. If the owner sells the cupcakes for $\$ 1.50$ apiece, how many cupcakes does she need to sell to start making profit? (Hint: Use the INTERSECTION function on a calculator to find this number.)

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Key Concepts

Linear Function

A linear function models a relationship where the change in one variable is constant with respect to another and can be represented by the equation y = mx + b. In cost analysis, it allows us to predict how total costs change with the number of items produced.

Fixed Cost

Fixed cost represents expenses that remain constant regardless of production volume. These costs, such as setup fees or facility rentals, do not vary with the number of units produced, serving as a baseline in cost functions.

Variable Cost

Variable cost is the expense that changes in direct proportion to the number of units produced. It includes costs like materials and labor that multiply as more products are made, forming the slope of the cost function when added to fixed costs.

Break-even Analysis

Break-even analysis determines the point at which total revenues equal total costs, meaning no profit is made but no losses are incurred either. It involves setting the revenue and cost functions equal to each other and solving for the number of units, indicating the minimum production needed to start making a profit.

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