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

Step-by-step explanation:

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.

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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.

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step-by-step explanation: