Question

A railroad tunnel is shaped like a semiellipse as shown below.

A semiellipse is shown on the coordinate plane with vertices on the x axis and one point of intersection with the positive y axis.

The height of the tunnel at the center is 58 ft and the vertical clearance must be 29 ft at a point 21 ft from the center. Find an equation for the ellipse.

Answer 1

@bohotness please help?:)

Answer 2

vertex= (0, 58)
h= 0
k= 58
ive got this so far do you know where I could go from here?:)

Answer 3

what is the standard form of an ellipse equation, itll give us something to start filling in

Answer 4

do you known it ?

Answer 5

(x - h)^2 = 4p(y - k), p ≠ 0

Answer 6

\[\frac{ (x - h)^2 }{ a^2 } + \frac{ (y-k)^2 }{ b^2 } = 1\]

Answer 7

a and b are your Semi-major axis lengths

Answer 8

You are given 2 points

Answer 9

(x-h^2)/a^2+(y-k)^2/b^2=1

Answer 10

OH WAIT sorry I gave you the formula for a parabola:/

Answer 11

its okay

Answer 12

Consider (0,0) the center of the elipse
(0,58) is on the ellipse
(21,29) is another point on the ellipse

Answer 13

ok well I know one of the points is (0, 58) and I guess I could assume that another point is (0, -58)...

Answer 14

and the h,k parts are 0 since we want to center it.

the under y part is the height at the center, squared.

Answer 15

all thats left to do is determine the value of a. any ideas?

Answer 16

right so 58 is your a or b value

Answer 17

semi-major axis length , center to the parahelion

Answer 18

b is the height stated in the problem, it is the distance from center to the y vertex ...

Answer 19

when y=b, then b^2/b^2 = 1

Answer 20

so, h=k=0 to center it at the origin of the graph ... and b=58 as stated. any ideas to determine a?

Answer 21

you can figure out the remaining value, if a = 58, and (h,k) =(0,0)
Find the value for 'b'

Answer 22

well I though 58 was = a because its the major axis

Answer 23

we have reduced it to one equation with 1 unknown, and a known point.

Answer 24

it would be the 'b' under the Y term

Answer 25

use the point and solve for a

Answer 26

rewrite your standard equation but with h=k=0 and b=58

Answer 27

\[\frac{ 21^2 }{ a^2 } + \frac{ 29^2 }{ 58^2 }= 1\]

Answer 28

it a = 588?

Answer 29

It will be the Positive root answer, it is a distance

Answer 30

I got
\[a = 14\sqrt{3} \approx 24.25\]

Answer 31

YA that's what I got when I square rooted 588:) thanks so much:)

Answer 32

yw

Answer 33

yeah so a^2 = (14 root 3)^2

Answer 34

yw