Tour Company Problem: Revenue, Cost & Profit Functions
A. Express the weekly revenue R as a function of p.
B. Express the weekly cost C as a function of p.
C. Find the equation for the weekly profit P as a function of p.
What This Problem Teaches
- How to build linear functions from real business scenarios
- Understanding the difference between fixed costs and variable costs
- Connecting revenue, cost, and profit through algebraic expressions
- Translating business language into mathematical functions
- Setting up the foundation for break-even analysis and optimization
Let's Visualize the Business Model
| Component | Type | Amount | Function |
|---|---|---|---|
| Revenue per tourist | Variable | 750 pesos | 750p |
| Fixed expenses | Fixed | 30,000 pesos/week | 30,000 |
| Variable cost per tourist | Variable | 150 pesos | 150p |
| Net profit per tourist | Contribution | 600 pesos | 750p - 150p = 600p |
Solution: Method 1 — Direct Linear Modeling
Part A — Building the Revenue Function
Revenue is simply the total money collected from customers. Each tourist pays 750 pesos, and we have p tourists per week.
R(p) = 750pThis makes intuitive sense: if 20 tourists book, revenue is 750 × 20 = 15,000 pesos. If 100 tourists book, revenue is 750 × 100 = 75,000 pesos.
Part B — Building the Cost Function
Cost has two components: fixed expenses that don't change with the number of tourists, and variable costs that increase with each additional tourist.
Fixed costs: 30,000 pesos per week (rent, salaries, etc.)
Variable costs: 150 pesos per tourist (taxes, entrance fees)
C(p) = 30,000 + 150pFor example, with 50 tourists: C(50) = 30,000 + 150(50) = 30,000 + 7,500 = 37,500 pesos
Part C — Finding the Profit Function
Profit equals revenue minus total cost:
P(p) = R(p) - C(p)P(p) = 750p - (30,000 + 150p)P(p) = 750p - 30,000 - 150pP(p) = 600p - 30,000Solution: Method 2 — Contribution Margin Analysis
Business analysts often think in terms of "contribution margin" — how much each unit sold contributes toward covering fixed costs and generating profit.
Step 1 — Calculate Contribution per Tourist
Each tourist contributes: Revenue per tourist - Variable cost per tourist
Contribution per tourist = 750 - 150 = 600 pesosStep 2 — Build Functions Using Contribution Logic
Revenue: Same as before, R(p) = 750p
Cost: Same as before, C(p) = 30,000 + 150p
Step 3 — Express Profit Using Contribution Margin
Total profit = (Contribution per tourist × Number of tourists) - Fixed costs
P(p) = 600p - 30,000This approach immediately shows that each tourist generates 600 pesos toward covering the 30,000 peso fixed cost. The company needs at least 30,000 ÷ 600 = 50 tourists to break even.
A. Revenue: R(p) = 750p
B. Cost: C(p) = 30,000 + 150p
C. Profit: P(p) = 600p - 30,000
Verification
Let's test our functions with p = 80 tourists:
Revenue check:R(80) = 750(80) = 60,000 pesos ✓
Cost check:C(80) = 30,000 + 150(80) = 30,000 + 12,000 = 42,000 pesos ✓
Profit check:P(80) = 60,000 - 42,000 = 18,000 pesos
Using profit function:P(80) = 600(80) - 30,000 = 48,000 - 30,000 = 18,000 pesos ✓
The calculations match perfectly, confirming our functions are correct.
Does This Seem Reasonable?
Let's examine some key scenarios to see if our model behaves sensibly:
This makes sense: with zero tourists, the company loses exactly their fixed costs. At 50 tourists, they break even. Beyond that, each additional tourist adds 600 pesos to profit.
Watch Out For These Mistakes
✗ Mistake 1: Writing revenue as R(p) = 750 + p
This would mean revenue increases by only 1 peso per additional tourist! The price multiplies the quantity: R(p) = 750p.
✗ Mistake 2: Forgetting the fixed costs in the cost function
Writing C(p) = 150p ignores the 30,000 peso weekly overhead. Even with zero tourists, the company still pays rent, salaries, etc.
✗ Mistake 3: Adding instead of subtracting when finding profit
Writing P(p) = R(p) + C(p) would give total money flow, not profit. Profit is what's left after paying costs: P(p) = R(p) - C(p).
The Pattern Behind This
This problem follows the standard linear business model:
Revenue = (Price per unit) × (Quantity)Cost = Fixed costs + (Variable cost per unit) × (Quantity)Profit = Revenue - Cost = (Price - Variable cost) × (Quantity) - Fixed costsThe key insight is that profit is determined by two factors:
- Contribution margin:
(Price - Variable cost) = 750 - 150 = 600 pesos per tourist - Break-even point:
Fixed costs ÷ Contribution margin = 30,000 ÷ 600 = 50 tourists
This framework applies to any business selling a single product or service.
Where This Shows Up in Real Life
- Manufacturing: Production cost = setup costs + (material + labor) per unit
- Software subscription: Monthly cost = server infrastructure + customer support per user
- Event planning: Total cost = venue rental + catering cost per guest
- Ride sharing: Driver earnings = base rate + per-mile rate × distance
Four "What-If?" Problems
Break-even occurs when P(p) = 0, so we solve: 600p - 30,000 = 0
600p = 30,000p = 30,000 ÷ 600 = 50
At 50 tourists: Revenue = 750(50) = 37,500, Cost = 30,000 + 150(50) = 37,500
Profit = 37,500 - 37,500 = 0 ✓
Answer: 50 tourists needed to break even
We want P(p) = 18,000, so: 600p - 30,000 = 18,000
600p = 18,000 + 30,000600p = 48,000p = 48,000 ÷ 600 = 80
At 80 tourists: P(80) = 600(80) - 30,000 = 48,000 - 30,000 = 18,000 ✓
Answer: 80 tourists needed for 18,000 pesos profit
With 800 peso price: R(p) = 800p
With higher fixed costs: C(p) = 35,000 + 150p
P(p) = 800p - (35,000 + 150p) = 650p - 35,000
650p - 35,000 = 0p = 35,000 ÷ 650 ≈ 53.8 ≈ 54 tourists
Answer: P(p) = 650p - 35,000; break-even at 54 tourists
Revenue = x · 70, Cost = 30,000 + 150(70) = 40,500
Profit: 70x - 40,500 = 15,000
70x = 15,000 + 40,50070x = 55,500x = 55,500 ÷ 70 = 793.57
At 793.57 pesos per tourist with 70 tourists:
Revenue = 793.57 × 70 = 55,550, Cost = 40,500
Profit = 55,550 - 40,500 = 15,050 ≈ 15,000 ✓
Answer: Must charge about 794 pesos per tourist
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2026-08-12