The solutions to the seven ellipse exercises and problems:
1. Perimeter Calculation: Given:
- Semi-major axis (a) = 12 inches
- Semi-minor axis (b) = 8 inches
Using the three perimeter formulas:
a. P ≈ π(a + b) = π(12 inches + 8 inches) = π(20 inches) ≈ 62.83 inches
b. P ≈ π√[2(a² + b²)] = π√[2(12 inches)² + (8 inches)²] ≈ π√[288 square inches + 64 square inches] ≈ π√352 square inches ≈ 59.27 inches
c. P ≈ π[(3/2)(a + b) - √(ab)] = π[(3/2)(12 inches + 8 inches) - √(12 inches * 8 inches)] ≈ π[(3/2)(20 inches) - √96 square inches] ≈ π[(30 inches) - √96 square inches] ≈ π[30 inches - 9.8 inches] ≈ π[20.2 inches] ≈ 63.46 inches
2. Area Calculation: Given:
- Semi-major axis (a) = 5 meters
- Semi-minor axis (b) = 3 meters
Area of ellipse = πab = π * 5 meters * 3 meters = 15π square meters ≈ 47.12 square meters
3. Eccentricity Calculation: Given:
- Semi-major axis (a) = 10 feet
- Semi-minor axis (b) = 6 feet
Eccentricity (e) = √(1 - b²/a²) = √(1 - (6 feet)² / (10 feet)²) = √(1 - 36/100) = √(0.64) ≈ 0.8
4. Latus Rectum Calculation: Given:
- Semi-major axis (a) = 7 inches
- Semi-minor axis (b) = 5 inches
Latus Rectum (L) = 2b²/a = 2 * (5 inches)² / 7 inches ≈ 7.14 inches
5. Semi-Major Axis Calculation: Given:
- Area (A) = 100 square centimeters
- Semi-minor axis (b) = 4 centimeters
Semi-major axis (a) = A / (π * b) = 100 square centimeters / (π * 4 centimeters) ≈ 7.96 centimeters
6. Semi-Minor Axis Calculation: Given:
- Area (A) = 150 square meters
- Semi-major axis (a) = 10 meters
Semi-minor axis (b) = A / (π * a) = 150 square meters / (π * 10 meters) ≈ 4.77 meters
7. Real-Life Application: Given:
- Length of semi-major axis (a) = 200 meters
- Length of semi-minor axis (b) = 50 meters
Perimeter of the racetrack ≈ π(a + b) = π(200 meters + 50 meters) ≈ π(250 meters) ≈ 785.4 meters
So, the runner needs to cover approximately 785.4 meters when running around the elliptical racetrack.