A 30ft cable is suspended between the tops of two 20ft poles on the level ground. The lowest point of the cable is 5ft above the ground.What is the distance between the two poles?

Question

mathmate · Answer

Draw a diagram:
|dw:1437334323698:dw|
If the cable is 30' long, what distance separates the two poles?

campbell_st · Answer

I would do it this way 

|dw:1437334277446:dw|

the x axis is the directrix, so the focal length is 5 units

a standard form of the parabola that uses focal length and directrix is

$$(x - h) = 4a(y - k)$$

where (h, k) is the vertex and a is the focal length 

so you have h = 0, k = 5 and a = 20

so the equation becomes 

$$(x - 0)^2 = 4 \times 5 \times (y - 5)$$

or

$$x^2 = 20(y- 5)$$

so substitute the height of the pole, y = 20 into the equation and solve for x. 

this distance will be the perpendicular from the y-axis to the pole

when you get the answer, double it to get the distance between the poles