The cable of a uniformly loaded suspension bridge hangs in the form of a parabola. The roadway which is horizontal and 100 m long is supported by vertical wires attached to the cable, the longest wire being 30 m and the shortest being 6 m. Find the length of a supporting wire attached to the roadway 18 m from the middle.

ncert cbse 11th mathematics  conic sections miscellaneous

3. The cable of a uniformly loaded suspension bridge hangs in the form of a parabola.The roadway which is horizontal and 100 m long is supported by vertical wires attached to the cable, the longest wire being 30 m and the shortest being 6 m. Find the length of a supporting wire attached to the roadway 18 m from the
middle.

Chose the parabola as (x^2)  = 4ay (open upwards)

Because the shortest wire is 6m at the centre , the road is hanging 6m below the vertex of the parabola, 

so the 6m has to be subtracted for all data to get distance above x axis on the parabola

 100 m long road means 50m either side of vertex 

longest wire 30 m will at either end 

and 30-6 = 24 m will be above x axis

so (50,24) lies on (x^2)  = 4ay

so 50^2 = 4 a (24) 

a = 2500 / 96

(x^2)  = 4ay

changes to  (x^2)  = 4(2500/96)y 

at 18m use (18,y)  which lies  on (x^2)  = 4(2500/96)y 

so 18^2 = 4(2500/96)y 

y =96*(18^2) / [4*2500]

y = 31104 / 10000

y = 3.1104 m above x axis.

extra 6m has to be added to get  the required length of wire

so required length of wire = 6 +3.1104 = 9.1104 m

or 9.11 m approximately.

 

4. An arch is in the form of a semi-ellipse. It is 8 m wide and 2 m high at the centre.Find the height of the arch at a point 1.5 m from one end. 

8m wide means

 choose length of major axis  2a = 8m 

 or a =8/2 = 4m

semi ellipse , 2 m high at the centre

means length of semi minor axis , b =2

equation of ellipse is

[ (x^2) / (a^2) ] + [ (y^2) / (b^2)] = 1

[ (x^2) / (4^2) ] + [ (y^2) / (2^2)] = 1

change given distance 1.5m to that in terms of centre (0,0) 

since a = 4

4-1.5 = 2.5 m

(2.5,y) lies on

[ (x^2) / (4^2) ] + [ (y^2) / (2^2)] = 1

[ ((2.5)^2) / (4^2) ] + [ (y^2) / (2^2)] = 1

  [ (y^2) / (2^2)] = 1 -[ ((2.5)^2) / (4^2) ]

  [ (y^2) / (2^2)] = 9.75/16

y^2 = 4*(9.75/16)

y^2 = (9.75/4)

y = [sqrt(9.75)]  / 2

y = 3.122/4

y =1.56 m

 

 

 

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ncert cbse 11th mathematics chapter 11 conic section 

miscellaneous

 If a parabolic reflector is 20 cm in diameter and 5 cm deep, find the focus

solution

2. An arch is in the form of a parabola with its axis vertical. The arch is 10 m high
and 5 m wide at the base. How wide is it 2 m from the vertex of the parabola?

solution

 

3. The cable of a uniformly loaded suspension bridge hangs in the form of a parabola.The roadway which is horizontal and 100 m long is supported by vertical wires attached to the cable, the longest wire being 30 m and the shortest being 6 m. Find the length of a supporting wire attached to the roadway 18 m from the
middle.

solution  

4. An arch is in the form of a semi-ellipse. It is 8 m wide and 2 m high at the centre.Find the height of the arch at a point 1.5 m from one end. 

solution 

 

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