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