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HELP PLEASE

A railroad tunnel is shaped like a semi ellipse, as shown below.



The height of the tunnel at the center is 27 ft, and the vertical clearance must be 9 ft at a point 24 ft from the center. Find an equation for the ellipse.

HELP PLEASE A railroad tunnel is shaped like a semi ellipse as shown below The height of the tunnel at the center is 27 ft and the vertical clearance must be 9 class=

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jcherry99
Remark
The general equation for an ellipse is
[tex] \frac{x^2}{a^2}+ \frac{y^2}{b^2}=1 [/tex] 

At the center, the the point is 0,27

Solving for the 
[tex] \frac{0}{a^2} + \frac{27^2}{b^2} = 1 [/tex]
or b^2= 27^2 * 1
Taking the square root of both sides 
b = 27

Now we need to use one more piece of information
The second point is (24,9) what that means is at the point x = 24, y = 9

Solving for that 
[tex]\frac{24^2}{a^2}+\frac{9^2}{27^2}=1[/tex]

576/a^2 + 81/729 = 1    Subtract 81/729 from both sides
576/a^2 = 1 - 81/729 = 576/a^2 = (729 - 81)/729 = 648 /729

576/a^2 = 648/729     Cross multiply

648 a^2 = 729 * 576   Divide by 648
a^2 = 648            

Full equation for the ellipse and answer is

[tex] \frac{x^2}{648}+ \frac{y^2}{729} = 1 [/tex]


 

gustanika
General equation for an ellipse with the center (0,0) could be determined using
[tex]\boxed{ \dfrac{x^{2}}{a^{2}}+ \dfrac{y^{2}}{b^{2}}=1}[/tex]
with stands for the length of the horizontal radius and b stands for the length of vertical radius

From the question above, we could draw informations:
b = 27 ft
one of the points on ellipse (x,y) = (24,9)

plug the numbers into the equation
[tex]\dfrac{x^{2}}{a^{2}}+ \dfrac{y^{2}}{b^{2}}=1[/tex]
[tex]\dfrac{24^{2}}{a^{2}}+ \dfrac{9^{2}}{27^{2}}=1[/tex]
[tex]\dfrac{576}{a^{2}}+ \dfrac{81}{729}=1[/tex]
[tex]\dfrac{576}{a^{2}}+ \dfrac{1}{9}=1[/tex]
solve for a²
[tex]\dfrac{576}{a^{2}} =1-\dfrac{1}{9}[/tex]
[tex]\dfrac{576}{a^{2}} =\dfrac{8}{9}[/tex]
[tex]a^{2} =\dfrac{9}{8} \times 576[/tex]
a² = 648

Input the value of a² to the equation
[tex]\boxed{ \dfrac{x^{2}}{648}+ \dfrac{y^{2}}{729}=1}[/tex]
This is the equation of the ellipse

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