In a suspension bridge, the shape of the suspension cables is parabolic. The bridge shown in the figure has towers that are 400 m apart, and the lowest point of the suspension cables is 100 m below the top of the towers. Find the equation of the parabolic part of the cables, placing the origin of the coordinate system at the lowest point of the cable.
NOTE: This equation is used to find the length of the cable needed in the construction of the bridge.
a. y^2=1,600x
b. x^2=-400y
c. x^2=400y
d. x^2=800y
e. x^2=1,600y
Find the standard form of the equation of the hyperbola with the given characteristics.
foci: (±4,0) asymptotes: y= ±5x
a. x^2/16 - y^2/25 = 1
b. x^2/8/13 - y^2/200/13 = 1
c. y^2/8/13 - x^2/200/13 = 1
d. y^2/16 - x^2/25 = 1
e. x^2/200/13 - y^2/8/13 = 1
+128475 We have the following points on the parabola (0,0), (200, 100), (-200, 100)
The vertex is the lowest point on the cable....this is located at (0, 0 ) and the parbola opens upward......so the form will be
y = ax^2
Since the point (200,100) is on the parbola, we can sove for "a" thusly
100 = a(200)^2
100 = a (40000)
a = 100/40000 = 1/400
So we have
y = (1/400)x^2 multiply both sides by 400
400y = x^2 → " c " is correct
+128475 Find the standard form of the equation of the hyperbola with the given characteristics.
foci: (±4,0) asymptotes: y= ±5x
We can answer this by process of elimination......
The center of the hyperbola is (0,0) and the focus lies on the x axis so the hyperbola opens left/right.......so the form will be
x^2 / a^2 - y^2 / b^2 = 1
Then, obviously, answers c and d are eliminated
The equation of one of the asymptotes will be [ +b/a ] x = 5x
Since b/a = 5 then [b/a]^2 = 25 → b^2/a^2 = 25
But in answer "a," b^2 = 25 and a^2 = 16....then b^2 / a^2 = 25/16 and this is < 25 so "a" is eliminated
And "e" is eliminated because a^2 > b^2....so b^2/a^2 < 1
So......answer "b" is correct
