SOLUTION: what is the area of ellipse with an aspect ratio of two and total perimeter of 25 inches?

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Question 1090034: what is the area of ellipse with an aspect ratio of two and total perimeter of 25 inches?
Answer by MathLover1(20850) About Me  (Show Source):
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what is the area of ellipse with an aspect ratio of two and total perimeter of 25 inches?

For an ellipse, the aspect ratio denotes the ratio of the major axis a to the minor axis b.

a and b are measured from the center, so they are like "radius" measures
given: a%2Fb=2 => a=2b
and perimeter P=25 inches
an approximation within about 5% of the true value, so long as a is not more than 3 times longer than b (in other words, the ellipse is not too "squashed"):

P=2pi%2Asqrt%28%28a%5E2%2Bb%5E2%29%2F2%29
25=2pi%2Asqrt%28%28%282b%29%5E2%2Bb%5E2%29%2F2%29.............solve for b
b+=+-%285+sqrt%285%2F2%29%29%2Fpi
b+≈ -2.51773699057992

a=2%2A%28-5sqrt%285%2F2%29%29%2Fpi+=+%28-10sqrt%285%2F2%29%29%2Fpi
a ≈ +-5.032921210448
Ellipse area is:
Area+=ab%2Api
Area+=%28-10+sqrt%285%2F2%29%29%2Fpi%2A%28-5+sqrt%285%2F2%29%29%2Fpi%29%2Api
Area+=%28-10+sqrt%285%2F2%29%29%2Fpi%2A%28-5sqrt%285%2F2%29%29
Area+=125%2Fpi
Area=39.8 in^2
check equation:

or
x%5E2%2F%28-5.032921210448%29%5E2+%2B+y%5E2%2F%28-2.51773699057992%29%5E2=1
it will give you:
area | 39.808917197
perimeter | 24.383816643 which is an approximation within about 5% of the true value