Asurveyor finds that the cross-section of the bottom of a
circular pond has the shape of a parabola. the pond is 24 feet
in diameter. the middle of the pond is the deepest part at
8 feet deep. at a point 2 feet from the shore the water is 3 feet
deep. how deep is the pond at a point 6 feet from the shore?
6 feet deep 6 feet from shore
A parabola of width 2 and depth 1 can be written as ...
y = x² -1
By scaling horizontally by a factor of 12 and vertically by a factor of 8, we can make it correspond to the cross section of the pond: width 24 and depth 8 at the center. This function will have its deepest point at x=0.
y = 8((x/12)² -1)
To move the deepest point to x=12, we can add a horizontal translation. That gives ...
y = 8(((x -12)/12)² -1)
A graph of this function is attached.
Unfortunately, the parabola matching the depth and width given will not match the depth at x=2. Rather, the depth is 3 feet at 12-√90 ≈ 2.513 feet from shore.
6 feet from shore, the depth is ...
y = 8(((6 -12)/12)² -1) = 8(-3/4) = -6
The pond is 6 feet deep 6 feet from shore.
circular pond has the shape of a par...