What is the estimated perimeter of an ellipse if the major axis has a length of 15 ft and the minor axis has a length of 7.5 ft?
you might start here:
http://www.mathsisfun.com/geometry/ellipse-perimeter.html
To find the estimated perimeter of an ellipse, we can use an approximation formula known as Ramanujan's second approximation for the circumference of an ellipse:
C ≈ π * (a + b) * (1 + (3 * h^2) / (10 + √(4 - 3 * h^2)))
Where:
- C is the estimated circumference (perimeter) of the ellipse,
- π is a mathematical constant approximately equal to 3.14159,
- a and b are the lengths of the major and minor axes respectively,
- h is the difference between the major and minor axes (h = (a - b) / (a + b)).
Given the major axis length (a) of 15 ft and the minor axis length (b) of 7.5 ft, we can calculate h as follows:
h = (a - b) / (a + b)
h = (15 - 7.5) / (15 + 7.5)
h = 7.5 / 22.5
h = 1 / 3
Substituting the values into the formula, we have:
C ≈ π * (15 + 7.5) * (1 + (3 * (1/3)^2) / (10 + √(4 - 3 * (1/3)^2)))
Simplifying further:
C ≈ π * 22.5 * (1 + 1/10)
C ≈ π * 22.5 * (11/10)
C ≈ π * 247.5 / 10
C ≈ 77.75π
Therefore, the estimated perimeter of the ellipse is approximately 77.75π feet.