the perimeter of an ellipse with an area equal to 12pi square units if the non opposite vertices are 5units apart
http://www.mathwarehouse.com/ellipse/area-of-ellipse.php
What a nasty question.
the area of the ellipse is abπ , where a and b are each 1/2 of the axes
so abπ = 12π --->ab = 6 or b = 6/a
let A(a,0) and B(0,b) be the two adjacent vertices
given AB = 5
√(a^2 + b^2) = 5
a^2 + b^2 = 25
a^2 + 36/a^2 = 25
a^4 + 36 = 25a^2
a^4 - 25a^2 + 36 = 0
a^2 = (25 ± √481)/2 = appr 23.4658 or appr 1.53414
a = 4.844157.. or a = 1.2386... ignoring the 2 negative values
then b = 6/a
if a = 4.844148, then b = 1.2386
if a = 1.2386 , then a = 4.844157
ahhh, summetrical answers
So we basically have the same ellipse, one horizontal, one vertical
The perimeter of an ellipse is one of the most challenging questions dealing with the ellipse.
It requires Calculus and finding the length of a curve.
Here is a link that has some approximation methods
http://www.mathsisfun.com/geometry/ellipse-perimeter.html
Here is an applet that let's you find all properties of an ellipse
http://www.cleavebooks.co.uk/scol/callipse.htm
enter the 2a= 9.6883
enter 2b = 2.47721..
and press calculate to get a perimeter of appr. 20.87
22.21
To find the perimeter of an ellipse, we need to determine the lengths of its major and minor axes.
Let's start by finding the length of the major axis. Since the non-opposite vertices are 5 units apart, we know that this distance represents the length of the major axis. Let's call this length 2a.
Therefore, a = 5/2 = 2.5 units.
Next, we need to find the length of the minor axis. The area of an ellipse is given by the formula: A = πab, where a and b are the lengths of the major and minor axes, respectively.
In this case, we have A = 12π square units. Plugging in the value for a, we can solve for b:
12π = π * 2.5 * b
12 = 2.5b
b = 12/2.5
b = 4.8 units
Now that we have the lengths of both the major and minor axes, we can calculate the perimeter of the ellipse using the formula: P = 2π √((a^2 + b^2) / 2).
Plugging in the values:
P = 2π √((2.5^2 + 4.8^2) / 2)
P = 2π √((6.25 + 23.04) / 2)
P = 2π √(29.29 / 2)
P = 2π √14.64
P = 2π * 3.83
P ≈ 24.06 units
Therefore, the perimeter of the ellipse is approximately 24.06 units.