SOLUTION: The City�s Parks and Recreation Department has called the Algebra Society to help them design the new gardens that they will be developing in the west end of town. For the gardens,

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Question 470340: The City�s Parks and Recreation Department has called the Algebra Society to help them design the new gardens that they will be developing in the west end of town. For the gardens, they have purchased a rectangular lot that is 42 feet wide and 87 feet long. The most important aspect of the new garden will be an elliptical walkway. In the center of the ellipse will be a grand fountain, and two smaller fountains are planned at the foci of the ellipse. The radius of the large fountain will be twice the radius of the smaller fountains. The centers of the small fountains will be the foci of the inner ellipse. The edge of the smaller fountains will be tangent(i.e. will touch) to the elliptical walkway. The Department provided the following sketch as a guide. The elliptical walkway will be 3 feet wide and tangent to a 3-foot walkway around the perimeter of the park.

Help the Department by finding the radius of the Grand Fountain and the radius of each smaller fountain.
Answer by ccs2011(207) (Show Source):
You can put this solution on YOUR website!
Let 2a be length of major axis of ellipse.
Let 2b be length of minor axis of ellipse.
At either end of these axis, there are 2 tangent walkways each 3-feet wide
Six feet on each end which makes 12 feet. Subtract 12 from length and width of park to find dimensions of ellipse.
2a = 87 - 12 = 75 --> a = 37.5
2b = 42 - 12 = 30 --> b = 15
The foci are located length c from center, where

The radius of small fountain is distance from focus point to edge of ellipse.
r = a - c
r = 37.5 - 34.369 = 3.13
Radius of grand fountain is twice that of smaller fountain
R = 2*r = 2*3.13 = 6.26
Therefore, the desired radii are 3.13 ft. and 6.26 ft.