Problem 105 - Calculus Volume 1

Question

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.)

Step-by-Step Solution

Answer

a. Linear function: \( C(x) = 0.75x + 125 \) b. Cost for 160 cupcakes: 245 dollars. c. Sell at least 167 cupcakes to make a profit.

Step 1: Define the cost function

The total cost to make the cupcakes includes a fixed setup cost and a variable cost per cupcake made. Let \( C \) be the total cost and \( x \) be the number of cupcakes made.- Fixed cost = \( 125 \ \text{dollars} \)- Variable cost per cupcake = \( 0.75 \ \text{dollars} \)The linear function for the total cost \( C \) is:\[ C(x) = 0.75x + 125 \]

Step 2: Calculate the cost for 160 cupcakes

Substitute \( x = 160 \) into the cost function \( C(x) = 0.75x + 125 \) to find the total cost for 160 cupcakes:\[C(160) = 0.75(160) + 125\]Calculate the values:\[C(160) = 120 + 125 = 245\]Therefore, the cost to bake 160 cupcakes is \( 245 \ \text{dollars} \).

Step 3: Establish profit condition

Profit occurs when the revenue from selling cupcakes exceeds the cost. Let each cupcake be sold for \( 1.50 \ \text{dollars} \), giving the revenue function:\[ R(x) = 1.50x \]To find when the bakery breaks even, set the revenue equal to the cost:\[ 1.50x = 0.75x + 125 \]

Step 4: Solve for break-even point

Solve the equation from Step 3 for \( x \):\[1.50x = 0.75x + 125\]Subtract \( 0.75x \) from both sides:\[0.75x = 125\]Divide by \( 0.75 \) to find \( x \):\[x = \frac{125}{0.75} = 166.67\]Since \( x \) represents the number of cupcakes, round up to the nearest whole number: \( x = 167 \). The owner needs to sell 167 cupcakes to start making a profit.

Key Concepts

Understanding the Cost Function

The cost function helps us determine the total expenses involved in producing a certain number of goods. In this case, the bakery aims to understand its total cost in making cupcakes for a festival. The concept of a cost function involves two primary types of costs: fixed and variable.

• Fixed cost: This is a one-time expense, not affected by the number of items produced. For the bakery, the fixed cost is the setup fee of \\(125 at the festival.
• Variable cost: This cost changes in direct proportion to the number of products manufactured. For each cupcake made, the bakery incurs a variable cost of \\)0.75.

To formulate the cost function, these costs are combined: \[ C(x) = 0.75x + 125 \]Here, \( C(x) \) is the total cost, and \( x \) is the number of cupcakes produced. This linear function reflects how the bakery's total costs grow as more cupcakes are baked.

Discovering the Break-Even Point

A break-even point is crucial for businesses, indicating the stage where total revenue generated equals total expenses, resulting in no profit but no loss either. For the bakery, determining the break-even point involves understanding when revenue from cupcake sales meets the baking costs.

• Revenue function: Given each cupcake is sold at \$1.50, the revenue function is defined as: \[ R(x) = 1.50x \]
• Equating revenue to cost: To find the break-even point, set the revenue equation equal to the cost function: \[ 1.50x = 0.75x + 125 \]

By solving this equation, we determine the number of cupcakes needed to break even. Reduce the equation by subtracting \( 0.75x \) from both sides, yielding:\[ 0.75x = 125 \]This simplifies to:\[ x = \frac{125}{0.75} = 166.67 \]Then, rounding up as the bakery can't sell a fraction of a cupcake, the owner must sell 167 cupcakes to reach break-even.

Exploring Revenue Calculation

Revenue is the income generated from selling goods or services. For the bakery, revenue calculation begins by considering the selling price per unit and the total units sold.

• Per cupcake revenue: Each cupcake is sold for \$1.50. Thus, for \( x \) cupcakes, the revenue is:\[ R(x) = 1.50x \]

This simple linear relationship shows that the revenue directly increases as the number of cupcakes sold increases. Understanding this helps the bakery plan its sales strategy, as maximizing revenue involves not only selling more cupcakes but also potentially optimizing pricing strategies.