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

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

amistre64 · Answer

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

anonymous · Answer

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

amistre64 · Answer

good, and the h,k parts are 0 since we want to center it.

the under y part is the height at the center, squared.

amistre64 · Answer

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

anonymous · Answer

a is the y-value?

amistre64 · Answer

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

amistre64 · Answer

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

anonymous · Answer

ok

amistre64 · Answer

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

amistre64 · Answer

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

amistre64 · Answer

use the point and solve for a

anonymous · Answer

what is the reduced equation again?

amistre64 · Answer

i hve no mouse to copy and paste with.  rewrite your standard equation but with h=k=0 and b=58

anonymous · Answer

(x-h)^2/a^2+(y-k)^2/b^2=1
(x-0)^/a^2+(y-0)^2/58^2
x^2/a^2+y^2/3364=1

amistre64 · Answer

good, now we need the point (x=21,y=29) if memory serves, so input those and solve for a

anonymous · Answer

(21)^2/a^2+(29)^2/3364=1
441/a^2+841/3364=1

amistre64 · Answer

so far so good, the rest is basic algebra ... subtract, multiply, and divide as needed

amistre64 · Answer

y reduces to 1/4 to help out lol

amistre64 · Answer

the y parts reduce to 1/4 that is .... i get lazy typing at times

anonymous · Answer

441/a^2+1/4=1
a^2=558
a=24.248

amistre64 · Answer

14 sqrt3  but yeah

amistre64 · Answer

it should suffice for a^2 = 588 since a^2 is part of the formula

anonymous · Answer

So, the equation is:$$x^2/14\sqrt3+1/4=1$$

amistre64 · Answer

no, x^2/a^2 + y^2/b^2 = 1  is the formula.

we simply used it to find a^2, and b^2 is given as 58^2

anonymous · Answer

Ok, so it is x^2/14sqrt3+y^2/58^2=1?

amistre64 · Answer

a^2 = 588 which is what the formula requires.  finding a was a pointless endeavor

amistre64 · Answer

x^2/5988 + y^2/58^2 = 1   is what we are looking for to satisfy the given information

amistre64 · Answer

pfft, me fat finngers ... x^2/588 + ...

anonymous · Answer

Ok,Thank You