# A string tied between two poles at a height 20 m sags and follows a path y = 4x^2 - 8x + 4. What is the distance between where it is tied and the lowest point.

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The string is tied between two poles with each end at a height 20 m. The string sags down and follows a path given by y = 4x^2 - 8x + 4.

The lowest point of the string is at a point x where y' = 0

y' = 8x - 8

8x - 8 = 0

=> x = 1

y for x = 1 is 4*1 - 8 + 4 = 4 - 8 + 4 = 0

The lowest point is at (1, 0)

It is tied at a point where y = 20.

4x^2 - 8x + 4 = 20

=> 4x^2 - 8x - 16 = 0

=> x^2 - 2x - 4 = 0

=> x1 = `2/2 + sqrt(4 + 16)/2`

=> x1 = `1 + sqrt 5`

x2 = `1 - sqrt 5`

The points where the string is tied is (1 + `sqrt 5` , 20) and (1 - `sqrt 5` , 20)

The distance between the point where it is tied and the lowest point is `sqrt((1 + sqrt 5 - 1)^2 + (0 - 20)^2)`

=> `sqrt(5 + 400)`

=> `9*sqrt 5`

**The distance between either of the points where the string is tied and its lowest point is 9*sqrt 5.**