Formula Application and Solutions
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Formula Application and Solutions
📚 Part 1: Formula Application
Formula: A = π r²
Substitute: A = π × (7 cm)²
Simplify: A = π × 49 cm² = 49π cm²
Numeric value: A ≈ 49 × 3.1416 = 153.94 cm²
Final answer: A ≈ 153.94 cm²
Check (reasonableness): For r = 7 cm, πr² is just under 4×49 = 196, and 154 cm² is consistent with π ≈ 3.14.
Formula: P = 2(l + w)
Substitute: P = 2(12 m + 5 m)
Simplify: P = 2(17 m) = 34 m
Final answer: P = 34 m
Check: Two lengths = 24 m and two widths = 10 m; 24 + 10 = 34 m, so the result is correct.
Formula: I = P × r × t
Substitute: I = $1,500 × 0.05 × 3
Simplify: I = $1,500 × 0.15 = $225
Total amount: A = P + I = $1,500 + $225 = $1,725
Final answers: Interest = $225; Total = $1,725
Check: 5% of $1,500 is $75 per year; over 3 years = $225, consistent with the calculation.
Formula: speed = distance ÷ time
Substitute: speed = 90 km ÷ 1.5 h
Simplify: speed = 60 km/h
Final answer: 60 km/h
Check: 60 km/h for 1.5 h gives 60 × 1.5 = 90 km, so result is reasonable.
✏️ Part 2: Applied Problem Solving
Formula: V = π r² h
Given: diameter = 10 cm → r = 5 cm; h = 20 cm
Substitute: V = π × (5 cm)² × 20 cm
Simplify: V = π × 25 cm² × 20 cm = 500π cm³
Numeric value: V ≈ 500 × 3.1416 = 1,570.8 cm³
Convert to litres: 1 L = 1,000 cm³ → V ≈ 1,570.8 ÷ 1,000 = 1.5708 L
Final answer: V ≈ 1.571 L (to 3 dp). Check: cylinder of radius 5 cm and height 20 cm should be a bit over 1.5 L, so result is reasonable.
Formula: c = √(a² + b²)
Substitute: c = √(6² + 8²) m
Simplify: c = √(36 + 64) m = √100 m = 10 m
Final answer: c = 10 m
Check: 6-8-10 is a common Pythagorean triple, so the result is correct and reasonable.
Formula: A = (a + b) ÷ 2 × h
Substitute: A = (8 m + 5 m) ÷ 2 × 4 m
Simplify: A = 13 m ÷ 2 × 4 m = 6.5 m × 4 m = 26 m²
Final answer: A = 26 m²
Check: Average base = 6.5 m; times height 4 m gives 26 m², consistent and reasonable.
Formula: A = P(1 + r)^t
Substitute: A = $2,000 × (1 + 0.04)²
Simplify: A = $2,000 × (1.04)² = $2,000 × 1.0816 = $2,163.20
Compound interest earned: I = A − P = $2,163.20 − $2,000 = $163.20
Final answers: Amount = $2,163.20; Interest = $163.20
Check: Approx 4% of $2,000 is $80 in year 1, then 4% of $2,080 ≈ $83.20 in year 2; total ≈ $163.20, matches the result.
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