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
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?
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.
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.
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