Math Problem Statement
The cost of producing xx t-shirts is modeled by C(x)=7x+200C(x)=7x+200, and the revenue from selling xx t-shirts is R(x)=12xR(x)=12x.
Create a system of equations to represent cost and revenue.
Graph both equations on the same axes, labeling the intersection point. What does the intersection represent?
Solution
To solve the problem, we start by creating a system of equations from the given information. Let:
- , the cost equation.
- , the revenue equation.
The system of equations is:
C(x) &= 7x + 200, \\ R(x) &= 12x. \end{aligned}$$ --- ### **Graphing the Equations** The cost equation $$C(x) = 7x + 200$$ is a straight line with: - Slope: 7, - Y-intercept: 200. The revenue equation $$R(x) = 12x$$ is a straight line with: - Slope: 12, - Y-intercept: 0. To graph both equations, we: 1. Plot $$C(x)$$ and $$R(x)$$ on the same axes. 2. Find their **intersection point** by solving $$C(x) = R(x)$$: $$7x + 200 = 12x.$$ Rearrange to isolate $$x$$: $$200 = 5x \implies x = 40.$$ Substitute $$x = 40$$ into either equation to find the corresponding $$y$$-value: $$C(40) = 7(40) + 200 = 480, \quad \text{so } (x, y) = (40, 480).$$ --- ### **Interpretation of the Intersection** The intersection point $$(40, 480)$$ represents the **break-even point**: - At $$x = 40$$, the revenue ($$480$$) equals the cost ($$480$$). - Producing and selling more than 40 t-shirts generates profit, while selling fewer results in a loss. --- #### I will now graph these equations. numpy as np import matplotlib.pyplot as plt # Define the cost and revenue functions x = np.linspace(0, 60, 500) # x values from 0 to 60 for a clear view cost = 7 * x + 200 # Cost function revenue = 12 * x # Revenue function # Intersection point x_intersect = 40 y_intersect = 480 # Plot the graph plt.figure(figsize=(10, 6)) plt.plot(x, cost, label="Cost: C(x) = 7x + 200", color="blue", linewidth=2) plt.plot(x, revenue, label="Revenue: R(x) = 12x", color="green", linestyle="--", linewidth=2) # Highlight the intersection point plt.scatter(x_intersect, y_intersect, color="red", zorder=5, label="Break-even Point (40, 480)") # Add labels, legend, and title plt.title("Cost vs. Revenue", fontsize=16) plt.xlabel("Number of T-Shirts Sold (x)", fontsize=14) plt.ylabel("Dollars ($)", fontsize=14) plt.axhline(0, color='black',linewidth=0.8, linestyle="--") plt.axvline(0, color='black',linewidth=0.8, linestyle="--") plt.legend(fontsize=12) plt.grid(alpha=0.3) # Show the plot plt.show()The graph above displays the cost equation $$C(x) = 7x + 200$$ (blue line) and the revenue equation $$R(x) = 12x$$ (green dashed line). The intersection point $$(40, 480)$$ is highlighted in red and represents the **break-even point**. ### Summary: - At $$x = 40$$, the cost and revenue are equal (\$480). - Producing and selling more than 40 t-shirts will result in profit; selling fewer will result in a loss. --- Would you like more details about solving or interpreting this graph? ### Related Questions: 1. How do we calculate profit for a given number of t-shirts? 2. What happens to the break-even point if the cost per t-shirt changes? 3. How can we determine the slope of each line and interpret it in this context? 4. What is the significance of the y-intercept in each equation? 5. How would adding a fixed fee to revenue affect the graph? ### Tip: Always label axes and critical points clearly on graphs to make their interpretation intuitive.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Algebra
Linear Equations
Graphing
Break-Even Analysis
Formulas
C(x) = 7x + 200
R(x) = 12x
Theorems
Intersection of lines represents equal values
Break-even theorem in economics
Suitable Grade Level
Grades 9-12