document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1165309>Question 1165309</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Finicky-Cat Gourmet Pet Food makes organic cat food and sells them in 2 kg bags. The company has annual fixed costs of $150,000 and variable costs of $4 per bag. Finicky-Cat sells each bag for $10. Production capacity is 50,000 bags per year.<BR>\n" );
document.write( "A.) Find the revenue and cost equation.<BR>\n" );
document.write( "B.) How many bags of cat food does the company have to sell to break even?<BR>\n" );
document.write( "I. What are total sales at the breakeven point?<BR>\n" );
document.write( "II. What is the percent capacity at the breakeven point?<BR>\n" );
document.write( "C.) The company anticipates that will be able to make and sell 40,000 bags of cat food this year. What will it cost to produce these 40,000 bags?<BR>\n" );
document.write( "D.)Graph the cost equation and the revenue equation on graph paper. Graph from 0-50000 bags.<BR>\n" );
document.write( "E.) Finicky-Cat wants to increase its selling price from the current $10 so that it could make a profit of $150,000 from selling 40,000 bags of cat food. What price must they charge? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #853105 by </B><A HREF=/tutors/aboutme.mpl?userid=CPhill><B>CPhill(2138)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=CPhill><IMG SRC=http://www.algebra.com/images/aboutme-small.gif BORDER=0 alt=\"About Me\" VALIGN=MIDDLE></A>&nbsp;<IMG SRC=http://www.algebra.com/cgi-bin/show-face.mpl?user=CPhill><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=853105\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> This is a classic Cost-Volume-Profit (CVP) analysis problem. I will first use the code interpreter to perform the necessary calculations for accuracy, and then present the step-by-step solution.\r<BR>\n" );
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document.write( "Here is the information provided:\r<BR>\n" );
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document.write( "  * **Fixed Costs ($F$):** $\$150,000$<BR>\n" );
document.write( "  * **Variable Cost per Bag ($V$):** $\$4$<BR>\n" );
document.write( "  * **Selling Price per Bag ($P$):** $\$10$<BR>\n" );
document.write( "  * **Production Capacity ($C$):** $50,000$ bags<BR>\n" );
document.write( "  * **Target Sales ($Q_T$):** $40,000$ bags (for parts C and E)<BR>\n" );
document.write( "  * **Target Profit ($\text{Profit}_T$):** $\$150,000$ (for part E)\r<BR>\n" );
document.write( "\n" );
document.write( "Let $Q$ be the number of bags of cat food.\r<BR>\n" );
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document.write( "```python?code_reference&code_event_index=2<BR>\n" );
document.write( "import numpy as np<BR>\n" );
document.write( "import matplotlib.pyplot as plt\r<BR>\n" );
document.write( "\n" );
document.write( "# --- Given Data ---<BR>\n" );
document.write( "F = 150000  # Fixed Costs<BR>\n" );
document.write( "V = 4       # Variable Cost per Bag<BR>\n" );
document.write( "P = 10      # Selling Price per Bag<BR>\n" );
document.write( "Capacity = 50000<BR>\n" );
document.write( "Q_anticipated = 40000<BR>\n" );
document.write( "Target_Profit = 150000\r<BR>\n" );
document.write( "\n" );
document.write( "# --- Part A: Revenue and Cost Equations ---<BR>\n" );
document.write( "# R(Q) = P * Q<BR>\n" );
document.write( "# C(Q) = F + V * Q\r<BR>\n" );
document.write( "\n" );
document.write( "# --- Part B: Breakeven Analysis ---<BR>\n" );
document.write( "# Breakeven occurs when R(Q_BE) = C(Q_BE)<BR>\n" );
document.write( "# P * Q_BE = F + V * Q_BE<BR>\n" );
document.write( "# Q_BE * (P - V) = F<BR>\n" );
document.write( "Q_BE = F / (P - V)\r<BR>\n" );
document.write( "\n" );
document.write( "# B.I: Total Sales at Breakeven<BR>\n" );
document.write( "Sales_BE = P * Q_BE\r<BR>\n" );
document.write( "\n" );
document.write( "# B.II: Percent Capacity at Breakeven<BR>\n" );
document.write( "Percent_Capacity_BE = (Q_BE / Capacity) * 100\r<BR>\n" );
document.write( "\n" );
document.write( "# --- Part C: Cost to produce 40,000 bags ---<BR>\n" );
document.write( "Cost_40K = F + V * Q_anticipated\r<BR>\n" );
document.write( "\n" );
document.write( "# --- Part E: Target Price Calculation ---<BR>\n" );
document.write( "# Target Profit = (P_new * Q_T) - (F + V * Q_T)<BR>\n" );
document.write( "# Target Profit + F + V * Q_T = P_new * Q_T<BR>\n" );
document.write( "# P_new = (Target Profit + F + V * Q_T) / Q_T<BR>\n" );
document.write( "P_new = (Target_Profit + F + V * Q_anticipated) / Q_anticipated\r<BR>\n" );
document.write( "\n" );
document.write( "# --- Part D: Graphing Data ---<BR>\n" );
document.write( "Q_range = np.arange(0, Capacity + 1, 10000) # x-axis range<BR>\n" );
document.write( "R_values = P * Q_range<BR>\n" );
document.write( "C_values = F + V * Q_range\r<BR>\n" );
document.write( "\n" );
document.write( "# Plotting<BR>\n" );
document.write( "plt.figure(figsize=(10, 6))<BR>\n" );
document.write( "plt.plot(Q_range, R_values, label='Revenue R(Q) = 10Q', color='green', linewidth=2)<BR>\n" );
document.write( "plt.plot(Q_range, C_values, label='Cost C(Q) = 150,000 + 4Q', color='red', linewidth=2)<BR>\n" );
document.write( "plt.scatter(Q_BE, Sales_BE, color='blue', zorder=5, label=f'Breakeven ({Q_BE:.0f}, {Sales_BE:.2f})')<BR>\n" );
document.write( "plt.axvline(Q_BE, color='blue', linestyle='--', linewidth=0.8)<BR>\n" );
document.write( "plt.axhline(Sales_BE, color='blue', linestyle='--', linewidth=0.8)\r<BR>\n" );
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document.write( "plt.title('Cost and Revenue Analysis (Dunder Mifflin)')<BR>\n" );
document.write( "plt.xlabel('Quantity (Q) in Bags')<BR>\n" );
document.write( "plt.ylabel('Amount in Dollars ($)')<BR>\n" );
document.write( "plt.legend()<BR>\n" );
document.write( "plt.grid(True, linestyle='--')<BR>\n" );
document.write( "plt.ticklabel_format( DISABLED_style= 'plain', axis='y') # Ensure y-axis labels are not in scientific notation<BR>\n" );
document.write( "plt.xlim(0, Capacity)<BR>\n" );
document.write( "plt.ylim(0, max(R_values.max(), C_values.max()) * 1.1)<BR>\n" );
document.write( "plt.savefig('cost_revenue_graph.png')\r<BR>\n" );
document.write( "\n" );
document.write( "print(f\"Q_BE: {Q_BE}\")<BR>\n" );
document.write( "print(f\"Sales_BE: {Sales_BE}\")<BR>\n" );
document.write( "print(f\"Percent_Capacity_BE: {Percent_Capacity_BE}\")<BR>\n" );
document.write( "print(f\"Cost_40K: {Cost_40K}\")<BR>\n" );
document.write( "print(f\"P_new: {P_new}\")\r<BR>\n" );
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document.write( "```\r<BR>\n" );
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document.write( "```text?code_stdout&code_event_index=2<BR>\n" );
document.write( "Q_BE: 25000.0<BR>\n" );
document.write( "Sales_BE: 250000.0<BR>\n" );
document.write( "Percent_Capacity_BE: 50.0<BR>\n" );
document.write( "Cost_40K: 310000<BR>\n" );
document.write( "P_new: 11.5\r<BR>\n" );
document.write( "\n" );
document.write( "```\r<BR>\n" );
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document.write( "[image-tag: code-generated-image-0-1765591087689332374]\r<BR>\n" );
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document.write( "Here are the solutions to the Cost-Volume-Profit analysis questions:\r<BR>\n" );
document.write( "\n" );
document.write( "## A.) Find the revenue and cost equation.\r<BR>\n" );
document.write( "\n" );
document.write( "Let $Q$ be the number of bags of cat food sold.\r<BR>\n" );
document.write( "\n" );
document.write( "### Revenue Equation ($R(Q)$)\r<BR>\n" );
document.write( "\n" );
document.write( "$$R(Q) = \text{Selling Price per Bag} \times Q$$<BR>\n" );
document.write( "$$\mathbf{R(Q) = 10Q}$$\r<BR>\n" );
document.write( "\n" );
document.write( "### Cost Equation ($C(Q)$)\r<BR>\n" );
document.write( "\n" );
document.write( "$$C(Q) = \text{Fixed Costs} + (\text{Variable Cost per Bag} \times Q)$$<BR>\n" );
document.write( "$$\mathbf{C(Q) = 150,000 + 4Q}$$\r<BR>\n" );
document.write( "\n" );
document.write( "-----\r<BR>\n" );
document.write( "\n" );
document.write( "## B.) How many bags of cat food does the company have to sell to break even?\r<BR>\n" );
document.write( "\n" );
document.write( "The breakeven point occurs when Total Revenue equals Total Cost ($R(Q) = C(Q)$).<BR>\n" );
document.write( "The contribution margin per bag is $P - V = \$10 - \$4 = \$6$.\r<BR>\n" );
document.write( "\n" );
document.write( "$$\text{Breakeven Quantity} (Q_{BE}) = \frac{\text{Fixed Costs}}{\text{Contribution Margin per Unit}}$$<BR>\n" );
document.write( "$$Q_{BE} = \frac{150,000}{10 - 4} = \frac{150,000}{6}$$<BR>\n" );
document.write( "$$\mathbf{Q_{BE} = 25,000 \text{ bags}}$$\r<BR>\n" );
document.write( "\n" );
document.write( "### I. What are total sales at the breakeven point?\r<BR>\n" );
document.write( "\n" );
document.write( "Total sales are the revenue at the breakeven quantity.<BR>\n" );
document.write( "$$\text{Sales}_{BE} = P \times Q_{BE} = 10 \times 25,000$$<BR>\n" );
document.write( "$$\mathbf{\text{Total Sales at Breakeven} = \$250,000}$$\r<BR>\n" );
document.write( "\n" );
document.write( "### II. What is the percent capacity at the breakeven point?\r<BR>\n" );
document.write( "\n" );
document.write( "$$\text{Percent Capacity} = \left(\frac{Q_{BE}}{\text{Production Capacity}}\right) \times 100\%$$<BR>\n" );
document.write( "$$\text{Percent Capacity} = \left(\frac{25,000}{50,000}\right) \times 100\% = 0.5 \times 100\%$$<BR>\n" );
document.write( "$$\mathbf{\text{Percent Capacity at Breakeven} = 50\%}$$\r<BR>\n" );
document.write( "\n" );
document.write( "-----\r<BR>\n" );
document.write( "\n" );
document.write( "## C.) The company anticipates that will be able to make and sell 40,000 bags of cat food this year. What will it cost to produce these 40,000 bags?\r<BR>\n" );
document.write( "\n" );
document.write( "Use the Cost Equation $C(Q) = 150,000 + 4Q$ with $Q = 40,000$.\r<BR>\n" );
document.write( "\n" );
document.write( "$$\text{Cost}_{40,000} = 150,000 + 4(40,000)$$<BR>\n" );
document.write( "$$\text{Cost}_{40,000} = 150,000 + 160,000$$<BR>\n" );
document.write( "$$\mathbf{\text{Total Cost} = \$310,000}$$\r<BR>\n" );
document.write( "\n" );
document.write( "-----\r<BR>\n" );
document.write( "\n" );
document.write( "## D.) Graph the cost equation and the revenue equation on graph paper. Graph from 0-50,000 bags.\r<BR>\n" );
document.write( "\n" );
document.write( "  * **Cost Line (Red):** Starts at the fixed cost of $\$150,000$ (the y-intercept) and increases with a slope of 4.<BR>\n" );
document.write( "      * At $Q=0$, Cost = $\$150,000$.<BR>\n" );
document.write( "      * At $Q=50,000$, Cost = $150,000 + 4(50,000) = \$350,000$.<BR>\n" );
document.write( "  * **Revenue Line (Green):** Starts at $\$0$ (the origin) and increases with a slope of 10.<BR>\n" );
document.write( "      * At $Q=50,000$, Revenue = $10(50,000) = \$500,000$.<BR>\n" );
document.write( "  * **Breakeven Point (Blue Dot):** Intersects at $Q=25,000$ bags and Revenue/Cost $=\$250,000$.\r<BR>\n" );
document.write( "\n" );
document.write( "-----\r<BR>\n" );
document.write( "\n" );
document.write( "## E.) Finicky-Cat wants to increase its selling price from the current $\$10$ so that it could make a profit of $\$150,000$ from selling 40,000 bags of cat food. What price must they charge?\r<BR>\n" );
document.write( "\n" );
document.write( "We use the Profit equation: $\text{Profit} = (\text{New Price} \times Q) - \text{Total Cost}$.\r<BR>\n" );
document.write( "\n" );
document.write( "  * **Target Profit ($\text{Profit}_T$):** $\$150,000$<BR>\n" );
document.write( "  * **Target Quantity ($Q_T$):** $40,000$ bags<BR>\n" );
document.write( "  * **Total Cost at $40,000$ bags ($C_{40,000}$):** $\$310,000$ (from part C)\r<BR>\n" );
document.write( "\n" );
document.write( "$$\text{Profit}_T = (\text{New Price} \times Q_T) - C_{40,000}$$<BR>\n" );
document.write( "$$150,000 = (\text{New Price} \times 40,000) - 310,000$$\r<BR>\n" );
document.write( "\n" );
document.write( "**Solve for New Price:**<BR>\n" );
document.write( "$$\text{New Price} \times 40,000 = 150,000 + 310,000$$<BR>\n" );
document.write( "$$\text{New Price} \times 40,000 = 460,000$$<BR>\n" );
document.write( "$$\text{New Price} = \frac{460,000}{40,000}$$<BR>\n" );
document.write( "$$\mathbf{\text{New Price} = \$11.50}$$ </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );