Apply mathematics to real life! Learn to model complex scenarios from science, engineering, and everyday situations using mathematical concepts and problem-solving strategies.
Problem: An engineer needs to design a bridge that can support 50,000 kg. The bridge will be 100 meters long with support pillars every 25 meters. Each pillar must support an equal share of the weight.
Step 1: Determine number of support pillars
Bridge length: 100m, Pillar spacing: 25m
Number of pillars = 100 ÷ 25 = 4 pillars
Step 2: Calculate weight per pillar
Total weight: 50,000 kg
Weight per pillar = 50,000 ÷ 4 = 12,500 kg
Step 3: Apply safety factor
Safety factor: 2.5 (standard for bridges)
Required capacity = 12,500 × 2.5 = 31,250 kg per pillar
Problem: A bacteria colony starts with 1000 cells and doubles every 3 hours. How many cells will there be after 12 hours? When will the population reach 1 million cells?
Step 1: Model the growth
Initial population: 1000 cells
Doubling time: 3 hours
Growth rate: 2^(t/3) where t is time in hours
Step 2: Calculate population after 12 hours
Population = 1000 × 2^(12/3) = 1000 × 2^4 = 1000 × 16 = 16,000 cells
Step 3: Find time for 1 million cells
1,000,000 = 1000 × 2^(t/3)
1000 = 2^(t/3)
t/3 = log₂(1000) ≈ 9.97
t ≈ 29.9 hours
Problem: A company produces widgets. The cost function is C(x) = 0.5x² + 10x + 100, and the revenue function is R(x) = 25x, where x is the number of widgets. Find the profit-maximizing production level.
Step 1: Find the profit function
P(x) = R(x) - C(x) = 25x - (0.5x² + 10x + 100)
P(x) = -0.5x² + 15x - 100
Step 2: Find the vertex (maximum point)
For P(x) = ax² + bx + c, vertex at x = -b/(2a)
x = -15/(2 × (-0.5)) = -15/(-1) = 15
Step 3: Calculate maximum profit
P(15) = -0.5(15)² + 15(15) - 100
P(15) = -112.5 + 225 - 100 = 12.5
Maximum profit: $12.50 at 15 widgets
Problem: A family wants to reduce their carbon footprint. They currently drive 15,000 miles/year in a car that gets 25 mpg. Gasoline produces 19.6 lbs CO₂ per gallon. If they switch to a hybrid that gets 50 mpg, how much CO₂ will they save annually?
Step 1: Calculate current CO₂ emissions
Current gallons used = 15,000 ÷ 25 = 600 gallons/year
Current CO₂ = 600 × 19.6 = 11,760 lbs/year
Step 2: Calculate new CO₂ emissions
New gallons used = 15,000 ÷ 50 = 300 gallons/year
New CO₂ = 300 × 19.6 = 5,880 lbs/year
Step 3: Calculate savings
CO₂ saved = 11,760 - 5,880 = 5,880 lbs/year
Percentage reduction = (5,880 ÷ 11,760) × 100% = 50%
Linear Programming:
Maximize/minimize objective function
Use: Resource allocation and production planning
Monte Carlo:
Random sampling for complex systems
Use: Risk analysis and system behavior prediction
Climate Modeling
Temperature and precipitation predictions
Population dynamics and ecosystem analysis
Market Analysis
Price prediction and trend analysis
Portfolio optimization and risk management
Urban Planning
Traffic flow and city development
Social network analysis and behavior modeling