Solve complex problems combining arithmetic operations with geometric calculations and measurements.
Problem: Sarah wants to plant a rectangular garden that is 12 feet long and 8 feet wide. She needs to put a fence around it and cover it with mulch. Fencing costs $5 per foot, and mulch costs $3 per square foot. How much will the project cost?
Step 1 - Study: Find total cost = fence cost + mulch cost
Step 2 - Think: Need perimeter for fence, area for mulch
Step 3 - Execute:
• Perimeter = 2(12 + 8) = 2(20) = 40 feet
• Area = 12 × 8 = 96 square feet
• Fence cost = 40 × $5 = $200
• Mulch cost = 96 × $3 = $288
• Total cost = $200 + $288 = $488
Step 4 - Prove: $488 seems reasonable for a 12×8 garden
Problem: A room is 15 feet long and 10 feet wide. The ceiling is 8 feet high. You want to paint the walls (not the ceiling or floor). Paint covers 100 square feet per gallon and costs $25 per gallon. How many gallons do you need and what's the total cost?
Step 1 - Study: Find wall area, then paint needed
Step 2 - Think: 4 walls, each 8 feet high
Step 3 - Execute:
• Two walls: 15 × 8 = 120 sq ft each
• Two walls: 10 × 8 = 80 sq ft each
• Total wall area = 2(120) + 2(80) = 240 + 160 = 400 sq ft
• Gallons needed = 400 ÷ 100 = 4 gallons
• Total cost = 4 × $25 = $100
Step 4 - Prove: 4 gallons for 400 sq ft makes sense
Problem: A rectangular pool is 20 meters long and 12 meters wide. A 2-meter wide walkway surrounds it. What's the area of the walkway?
Your solution:
Problem: A storage box is 3 feet long, 2 feet wide, and 4 feet tall. You want to fill it with smaller boxes that are 1 foot long, 1 foot wide, and 1 foot tall. How many small boxes can fit? If each small box costs $8, what's the total value?
Step 1 - Study: Find how many small boxes fit, then calculate value
Step 2 - Think: Volume of large box ÷ volume of small box
Step 3 - Execute:
• Large box volume = 3 × 2 × 4 = 24 cubic feet
• Small box volume = 1 × 1 × 1 = 1 cubic foot
• Number of small boxes = 24 ÷ 1 = 24 boxes
• Total value = 24 × $8 = $192
Step 4 - Prove: 24 boxes × $8 = $192 ✓
Problem: A rectangular water tank is 5 meters long, 3 meters wide, and 2 meters deep. It's currently 60% full. How many liters of water are in the tank? (1 cubic meter = 1000 liters)
Step 1 - Study: Find total volume, then 60% of that
Step 2 - Think: Volume × 0.6, then convert to liters
Step 3 - Execute:
• Total volume = 5 × 3 × 2 = 30 cubic meters
• Water volume = 30 × 0.6 = 18 cubic meters
• Convert to liters = 18 × 1000 = 18,000 liters
Step 4 - Prove: 18,000 liters seems reasonable for a large tank
Problem: A swimming pool is 25 meters long, 15 meters wide, and 2 meters deep. It costs $0.50 per cubic meter to fill with water. How much will it cost to fill the pool?
Your solution:
Problem: A school wants to build a playground with a rectangular area (30m × 20m) and a circular sandbox (radius 3m). The rectangular area needs grass at $2 per square meter, and the sandbox needs sand at $15 per cubic meter (depth 0.5m). What's the total cost?
Step 1 - Study: Calculate costs for both areas separately
Step 2 - Think: Rectangle area + circle area, different materials
Step 3 - Execute:
• Rectangle area = 30 × 20 = 600 sq m
• Rectangle cost = 600 × $2 = $1,200
• Circle area = π × 3² = 3.14 × 9 = 28.26 sq m
• Sand volume = 28.26 × 0.5 = 14.13 cubic m
• Sand cost = 14.13 × $15 = $211.95
• Total cost = $1,200 + $211.95 = $1,411.95
Step 4 - Prove: Check calculations and reasonableness
Problem: A room is 12 feet long, 10 feet wide, and 9 feet tall. You want to install baseboard around the room ($3 per foot) and paint the walls ($0.50 per square foot). What's the total cost?
Your solution:
A rectangular patio is 16 feet long and 12 feet wide. You want to put a 2-foot wide border of flowers around it. The patio costs $8 per square foot, and the flower border costs $3 per square foot. What's the total cost?
Your solution:
A storage room is 8 feet long, 6 feet wide, and 7 feet tall. You want to store boxes that are 2 feet long, 1.5 feet wide, and 1 foot tall. How many boxes can you store? If each box contains items worth $50, what's the total value?
Your solution:
A circular pool has a diameter of 20 feet and is 4 feet deep. The pool liner costs $12 per square foot, and filling it with water costs $0.10 per gallon. (1 cubic foot = 7.48 gallons) What's the total cost?
Your solution: