The arch of an underpass is a semi - ellipse 60ft wide and 20 ft high. find the clearance at the edge of a lane if the edge is 20ft from the middle.
Backofbook:
Don't I love ellipses.
So the ellipse is horizontal. Its major axis, 2a, is 60 ft. Its minor axis, 2b, is 2*20 ft.
If the center of the ellipse is the center of the lane, then the ellipse is
(x^2)/(a^2) +(y^2)/(b^2) = 1
(x^2)/(30^2) +(y^2)/(20^2) = 1 -----(i)
At the edge of the lane, or at x = 20 ft, what is the y there?
(20^2)/(30^2) +(y^2)/(20^2) = 1
Clear the fractions, multiply both sides by (30^2)(20^2),
(20^2)(20^2) +(y^2)(30^2) = (30^2)(20^2)
(30^2)(y^2) = (30^2)(20^2) -(20^2)(20^2)
(30^2)(y^2) = (20^2)(30^2 -20^2)
(30^2)(y^2) = (20^2)[(30+20)(30-20)]
(30^2)(y^2) = (20^2)(50*10)
(30^2)(y^2) = (20^2)(500)
(30^2)(y^2) = (20^2)(10^2)(5)
y^2 = [(20^2)(10^2)(5)] / (30^2)
Take the square roots of both sides,
y = [(20)(10)sqrt(5)] / 30
y = 200sqrt(5) / 30
y = (20/3)sqrt(5) ft. ---------------answer.
Hello, ^_^Engineer_Adam^_^!
ticbol provided an excellent solution . . . as usual.
. . I simplified the arithmetic/algebra.
This ellipse has: a = 30, b = 20The arch of an underpass is a semi-ellipse, 60ft wide and 20 ft high.
Find the clearance at the edge of a lane if the edge is 20ft from the middle.
Back of book: ft
The equation is: .x²/30² + y²/20² .= .1
. . . . . . . . . . . . . . . . . . . _______
And we have: . y .= .(2/3)√900 - x²
. . . . . . . . . . . . . . . . . . . ___ . . . . . . . . ._ . . . . . . . . ._
When x = 20: .y .= .(2/3)√500 .= .(2/3)10√5 .= . (20/3)√5
I'm still waiting for LaTeX to return . . .